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QuickMath will automatically answer the most common problems in algebra, equations and calculus faced by high-school and college students.

• The algebra section allows you to expand, factor or simplify virtually any expression you choose. It also has commands for splitting fractions into partial fractions, combining several fractions into one and cancelling common factors within a fraction.
• The equations section lets you solve an equation or system of equations. You can usually find the exact answer or, if necessary, a numerical answer to almost any accuracy you require.
• The inequalities section lets you solve an inequality or a system of inequalities for a single variable. You can also plot inequalities in two variables.
• The calculus section will carry out differentiation as well as definite and indefinite integration.
• The matrices section contains commands for the arithmetic manipulation of matrices.
• The graphs section contains commands for plotting equations and inequalities.
• The numbers section has a percentages command for explaining the most common types of percentage problems and a section for dealing with scientific notation.

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Enter your queries using plain English. To avoid ambiguous queries, make sure to use parentheses where necessary. Here are some examples illustrating how to formulate queries.

• find roots to quadratic x^2-7x+12
• plot inequality x^2-7x+12<=0
• solve {3x-5y==2,x+2y==-1}
• plot inequality 3x-5y>=2 and x+2y<=-1
• solve 3x^2-y^2==2 and x+2y^2==5
• plot 3x^2-y^2>=2 and x+2y^2<=5
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About solving equations

A value is said to be a root of a polynomial if ..

The largest exponent of appearing in is called the degree of . If has degree , then it is well known that there are roots, once one takes into account multiplicity. To understand what is meant by multiplicity, take, for example, . This polynomial is considered to have two roots, both equal to 3.

One learns about the "factor theorem," typically in a second course on algebra, as a way to find all roots that are rational numbers. One also learns how to find roots of all quadratic polynomials, using square roots (arising from the discriminant) when necessary. There are more advanced formulas for expressing roots of cubic and quartic polynomials, and also a number of numeric methods for approximating roots of arbitrary polynomials. These use methods from complex analysis as well as sophisticated numerical algorithms, and indeed, this is an area of ongoing research and development.

Systems of linear equations are often solved using Gaussian elimination or related methods. This too is typically encountered in secondary or college math curricula. More advanced methods are needed to find roots of simultaneous systems of nonlinear equations. Similar remarks hold for working with systems of inequalities: the linear case can be handled using methods covered in linear algebra courses, whereas higher-degree polynomial systems typically require more sophisticated computational tools.

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• Pre Algebra Order of Operations Factors & Primes Fractions Long Arithmetic Decimals Exponents & Radicals Ratios & Proportions Percent Modulo Mean, Median & Mode Scientific Notation Arithmetics
• Algebra Equations Inequalities System of Equations System of Inequalities Basic Operations Algebraic Properties Partial Fractions Polynomials Rational Expressions Sequences Power Sums Interval Notation Pi (Product) Notation Induction Logical Sets Word Problems
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• Trigonometry Identities Proving Identities Trig Equations Trig Inequalities Evaluate Functions Simplify
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• Finance Simple Interest Compound Interest Present Value Future Value
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• Powers of i
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• -x+3\gt 2x+1
• (x+5)(x-5)\gt 0
• 10^{1-x}=10^4
• \sqrt{3+x}=-2
• 6+11x+6x^2+x^3=0
• factor\:x^{2}-5x+6
• simplify\:\frac{2}{3}-\frac{3}{2}+\frac{1}{4}
• x+2y=2x-5,\:x-y=3
• How do you solve algebraic expressions?
• To solve an algebraic expression, simplify the expression by combining like terms, isolate the variable on one side of the equation by using inverse operations. Then, solve the equation by finding the value of the variable that makes the equation true.
• What are the basics of algebra?
• The basics of algebra are the commutative, associative, and distributive laws.
• What are the 3 rules of algebra?
• The basic rules of algebra are the commutative, associative, and distributive laws.
• What is the golden rule of algebra?
• The golden rule of algebra states Do unto one side of the equation what you do to others. Meaning, whatever operation is being used on one side of equation, the same will be used on the other side too.
• What are the 5 basic laws of algebra?
• The basic laws of algebra are the Commutative Law For Addition, Commutative Law For Multiplication, Associative Law For Addition, Associative Law For Multiplication, and the Distributive Law.

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• High School Math Solutions – Systems of Equations Calculator, Elimination A system of equations is a collection of two or more equations with the same set of variables. In this blog post,... Read More

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• Adding Using Long Addition
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• Converting Regular to Scientific Notation
• Arranging a List in Order
• Expanded Notation
• Prime or Composite
• Comparing Expressions
• Converting to a Percentage
• Finding the Additive Inverse
• Finding the Multiplicative Inverse
• Reducing Fractions
• Finding the Reciprocal
• Converting to a Decimal
• Converting to a Mixed Number
• Adding Fractions
• Subtracting Fractions
• Multiplying Fractions
• Dividing Fractions
• Converting Ratios to Fractions
• Converting Percents to Decimal
• Converting Percents to Fractions
• Converting the Percent Grade to Degree
• Converting the Degree to Percent Grade
• Finding the Area of a Rectangle
• Finding the Perimeter of a Rectangle
• Finding the Area of a Square
• Finding the Perimeter of a Square
• Finding the Area of a Circle
• Finding the Circumference of a Circle
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• Finding the Area of a Trapezoid
• Finding the Volume of a Box
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• Finding the Volume of a Cone
• Finding the Volume of a Pyramid
• Finding the Volume of a Sphere
• Finding the Surface Area of a Box
• Finding the Surface Area of a Cylinder
• Finding the Surface Area of a Cone
• Finding the Surface Area of a Pyramid
• Converting to a Fraction
• Simple Exponents
• Prime Factorizations
• Finding the Factors
• Simplifying Fractions
• Converting Grams to Kilograms
• Converting Grams to Pounds
• Converting Grams to Ounces
• Converting Feet to Inches
• Converting to Meters
• Converting Feet to Miles
• Converting Feet to Yards
• Converting to Feet
• Converting to Yards
• Converting Miles to Feet
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• Converting Miles to Yards
• Converting Kilometers to Miles
• Converting Kilometers to Meters
• Converting Meters to Feet
• Converting Meters to Inches
• Converting Ounces to Grams
• Converting Ounces to Pounds
• Converting Ounces to Tons
• Converting Pounds to Grams
• Converting Pounds to Ounces
• Converting Pounds to Tons
• Converting Yards to Feet
• Converting Yards to Millimeters
• Converting Yards to Inches
• Converting Yards to Miles
• Converting Yards to Meters
• Converting Fahrenheit to Celsius
• Converting Celsius to Fahrenheit
• Finding the Median
• Finding the Mean (Arithmetic)
• Finding the Mode
• Finding the Minimum
• Finding the Maximum
• Finding the Lower or First Quartile
• Finding the Upper or Third Quartile
• Finding the Five Number Summary
• Finding a Point's Quadrant
• Finding the Midpoint of a Line Segment
• Distance Formula
• Arithmetic Operations
• Combining Like Terms
• Determining if the Expression is a Polynomial
• Distributive Property
• Simplifying
• Multiplication
• Polynomial Addition
• Polynomial Subtraction
• Polynomial Multiplication
• Polynomial Division
• Simplifying Expressions
• Evaluate the Expression Using the Given Values
• Multiplying Polynomials Using FOIL
• Identifying Degree
• Operations on Polynomials
• Negative Exponents
• Evaluating Radicals
• Solving by Adding/Subtracting
• Solving by Multiplying/Dividing
• Solving Containing Decimals
• Solving for a Variable
• Solving Linear Equations
• Solving Linear Inequalities
• Finding the Quadratic Constant of Variation
• Converting the Percent Grade to Slope
• Converting the Slope to Percent Grade
• Finding Equations Using Slope-Intercept
• Finding the Slope
• Finding the y Intercept
• Calculating Slope and y-Intercept
• Rewriting in Slope-Intercept Form
• Finding Equations Using the Slope-Intercept Formula
• Finding Equations Using Two Points
• Finding a Perpendicular Line Containing a Given Point
• Finding a Parallel Line Containing a Given Point
• Finding a Parallel Line to the Given Line
• Finding a Perpendicular Line to the Given Line
• Finding Ordered Pair Solutions
• Using a Table of Values to Graph an Equation
• Finding the Equation Using Point-Slope Form
• Finding the Surface Area of a Sphere
• Solving by Graphing
• Finding the LCM of a List of Expressions
• Finding the LCD of a List of Expressions
• Determining if the Number is a Perfect Square
• Finding the Domain
• Evaluating the Difference Quotient
• Solving Using the Square Root Property
• Determining if True
• Finding the Holes in a Graph
• Finding the Common Factors
• Expand a Trinomial with the Trinomial Theorem
• Finding the Start Point Given the Mid and End Points
• Finding the End Point Given the Start and Mid Points
• Finding the Slope and y-Intercept
• Finding the Equation of the Parabola
• Finding the Average Rate of Change
• Finding the Slope of the Perpendicular Line to the Line Through the Two Points
• Rewriting Using Negative Exponents
• Synthetic Division
• Maximum Number of Real Roots/Zeros
• Finding All Possible Roots/Zeros (RRT)
• Finding All Roots with Rational Root Test (RRT)
• Finding the Remainder
• Finding the Remainder Using Long Polynomial Division
• Reordering the Polynomial in Ascending Order
• Reordering the Polynomial in Descending Order
• Finding the Leading Term
• Finding the Leading Coefficient
• Finding the Degree, Leading Term, and Leading Coefficient
• Finding the GCF of a Polynomial
• Factoring Out Greatest Common Factor (GCF)
• Identifying the Common Factors
• Cancelling the Common Factors
• Finding the LCM using GCF
• Finding the GCF
• Factoring Trinomials
• Trinomial Squares
• Factoring Using Any Method
• Factoring a Difference of Squares
• Factoring a Sum of Cubes
• Factoring by Grouping
• Factoring a Difference of Cubes
• Determine if an Expression is a Factor
• Determining if Factor Using Synthetic Division
• Find the Factors Using the Factor Theorem
• Determining if Polynomial is Prime
• Determining if the Polynomial is a Perfect Square
• Expand using the Binomial Theorem
• Factoring over the Complex Numbers
• Finding All Integers k Such That the Trinomial Can Be Factored
• Determining if Linear
• Rewriting in Standard Form
• Finding x and y Intercepts
• Finding Equations Using the Point Slope Formula
• Finding Equations Given Point and y-Intercept
• Finding the Constant Using Slope
• Finding the Slope of a Parallel Line
• Finding the Slope of a Perpendicular Line
• Simplifying Absolute Value Expressions
• Solving with Absolute Values
• Finding the Vertex for the Absolute Value
• Rewriting the Absolute Value as Piecewise
• Calculating the Square Root
• Simplifying Radical Expressions
• Rationalizing Radical Expressions
• Solving Radical Equations
• Rewriting with Rational (Fractional) Exponents
• Finding the Square Root End Point
• Operations on Rational Expressions
• Determining if the Point is a Solution
• Solving over the Interval
• Finding the Range
• Finding the Domain and Range
• Solving Rational Equations
• Adding Rational Expressions
• Subtracting Rational Expressions
• Multiplying Rational Expressions
• Finding the Equation Given the Roots
• Finding the Asymptotes
• Finding the Constant of Variation
• Finding the Equation of Variation
• Substitution Method
• Addition/Elimination Method
• Graphing Method
• Determining Parallel Lines
• Determining Perpendicular Lines
• Dependent, Independent, and Inconsistent Systems
• Finding the Intersection (and)
• Using the Simplex Method for Constraint Maximization
• Using the Simplex Method for Constraint Minimization
• Finding the Union (or)
• Finding the Equation with Real Coefficients
• Solving in Terms of the Arbitrary Variable
• Finding a Direct Variation Equation
• Finding the Slope for Every Equation
• Finding a Variable Using the Constant of Variation
• Quadratic Formula
• Solving by Factoring
• Solve by Completing the Square
• Finding the Perfect Square Trinomial
• Finding the Quadratic Equation Given the Solution Set
• Finding a,b, and c in the Standard Form
• Finding the Discriminant
• Finding the Zeros by Completing the Square
• Quadratic Inequalities
• Rational Inequalities
• Converting from Interval to Inequality
• Converting to Interval Notation
• Rewriting as a Single Interval
• Determining if the Relation is a Function
• Finding the Domain and Range of the Relation
• Finding the Inverse of the Relation
• Finding the Inverse
• Determining if One Relation is the Inverse of Another
• Determining if Surjective (Onto)
• Determining if Bijective (One-to-One)
• Determining if Injective (One to One)
• Rewriting as an Equation
• Rewriting as y=mx+b
• Solving Function Systems
• Find the Behavior (Leading Coefficient Test)
• Determining Odd and Even Functions
• Describing the Transformation
• Finding the Symmetry
• Arithmetic of Functions
• Domain of Composite Functions
• Finding Roots Using the Factor Theorem
• Determine if Injective (One to One)
• Determine if Surjective (Onto)
• Finding the Vertex
• Finding the Sum
• Finding the Difference
• Finding the Product
• Finding the Quotient
• Finding the Domain of the Sum of the Functions
• Finding the Domain of the Difference of the Functions
• Finding the Domain of the Product of the Functions
• Finding the Domain of the Quotient of the Functions
• Finding Roots (Zeros)
• Identifying Zeros and Their Multiplicities
• Finding the Bounds of the Zeros
• Proving a Root is on the Interval
• Finding Maximum Number of Real Roots
• Function Composition
• Rewriting as a Function
• Determining if a Function is Rational
• Determining if a Function is Proper or Improper
• Maximum/Minimum of Quadratic Functions
• Finding All Complex Number Solutions
• Rationalizing with Complex Conjugates
• Vector Arithmetic
• Finding the Complex Conjugate
• Finding the Magnitude of a Complex Number
• Simplifying Logarithmic Expressions
• Expanding Logarithmic Expressions
• Evaluating Logarithms
• Rewriting in Exponential Form
• Converting to Logarithmic Form
• Exponential Expressions
• Exponential Equations
• Converting to Radical Form
• Find the Nth Root of the Given Value
• Simplifying Matrices
• Finding the Variables
• Solving the System of Equations Using an Inverse Matrix
• Finding the Dimensions
• Multiplication by a Scalar
• Subtraction
• Finding the Determinant of the Resulting Matrix
• Finding the Inverse of the Resulting Matrix
• Finding the Identity Matrix
• Finding the Scalar multiplied by the Identity Matrix
• Simplifying the Matrix Operation
• Finding the Determinant of a 2x2 Matrix
• Finding the Determinant of a 3x3 Matrix
• Finding the Determinant of Large Matrices
• Inverse of a 2x2 Matrix
• Inverse of an nxn Matrix
• Finding Reduced Row Echelon Form
• Finding the Transpose
• Finding the Adjoint
• Finding the Cofactor Matrix
• Finding the Pivot Positions and Pivot Columns
• Finding the Basis and Dimension for the Row Space of the Matrix
• Finding the Basis and Dimension for the Column Space of the Matrix
• Finding the LU Decomposition of a Matrix
• Identifying Conic Sections
• Identifying Circles
• Finding a Circle Using the Center and Another Point
• Finding a Circle by the Diameter End Points
• Finding the Parabola Equation Using the Vertex and Another Point
• Finding the Properties of the Parabola
• Finding the Vertex Form of the Parabola
• Finding the Vertex Form of an Ellipse
• Finding the Vertex Form of a Circle
• Finding the Vertex Form of a Hyperbola
• Finding the Standard Form of a Parabola
• Finding the Expanded Form of an Ellipse
• Finding the Expanded Form of a Circle
• Finding the Expanded Form of a Hyperbola
• Vector Addition
• Vector Subtraction
• Vector Multiplication by a Scalar
• Finding the Length
• Finding the Position Vector
• Determining Column Spaces
• Finding an Orthonormal Basis by Gram-Schmidt Method
• Rewrite the System as a Vector Equality
• Finding the Rank
• Finding the Nullity
• Finding the Distance
• Finding the Plane Parallel to a Line Given four 3d Points
• Finding the Intersection of the Line Perpendicular to Plane 1 Through the Origin and Plane 2
• Finding the Eigenvalues
• Finding the Characteristic Equation
• Finding the Eigenvectors/Eigenspace of a Matrix
• Proving a Transformation is Linear
• Finding the Kernel of a Transformation
• Projecting Using a Transformation
• Finding the Pre-Image
• Finding the Intersection of Sets
• Finding the Union of Number Sets
• Determining if a Set is a Subset of Another Set
• Determining if Two Sets are Mutually Exclusive
• Finding the Set Complement of Two Sets
• Finding the Power Set
• Finding the Cardinality
• Finding the Cartesian Product of Two Sets
• Determining if a Set is a Proper Subset of Another Set
• Finding the Function Rule
• Finding the Square or Rectangle Area Given Four Points
• Finding the Square or Rectangle Perimeter Given Four Points
• Finding the Square or Rectangle Area Given Three Points
• Finding the Square or Rectangle Perimeter Given Three Points
• Finding the Equation of a Circle
• Finding the Equation of a Hyperbola
• Finding the Equation of an Ellipse
• Partial Fraction Decomposition
• Finding an Angle Using another Angle
• Pythagorean Theorem
• Finding the Sine
• Finding the Cosine
• Finding the Tangent
• Finding the Trig Value
• Converting to Degrees, Minutes, and Seconds
• Finding Trig Functions Using Identities
• Finding Trig Functions Using the Right Triangle
• Converting Radians to Degrees
• Converting Degrees to Radians
• Finding a Reference Angle
• Finding a Supplement
• Finding a Complement
• Converting RPM to Radians per Second
• Finding the Quadrant of the Angle
• Graphing Sine & Cosine Functions
• Graphing Other Trigonometric Functions
• Amplitude, Period, and Phase Shift
• Finding the Other Trig Values in a Quadrant
• Finding the Exact Value
• Finding the Value Using the Unit Circle
• Expanding Trigonometric Expressions
• Expanding Using Double-Angle Formulas
• Expanding Using Triple-Angle Formulas
• Expanding Using Sum/Difference Formulas
• Simplify Using Pythagorean Identities
• Simplify by Converting to Sine/Cosine
• Inverting Trigonometric Expressions
• Finding the Trig Value of an Angle
• Expanding Using De Moivre's Theorem
• Verifying Trigonometric Identities
• Using Fundamental Identities
• Solving Standard Angle Equations
• Complex Trigonometric Equations
• Solving the Triangle
• Find the Roots of a Complex Number
• Complex Operations
• Trigonometric Form of a Complex Number
• Converting to Polar Coordinates
• Identifying and Graphing Circles
• Identifying and Graphing Limacons
• Identifying and Graphing Roses
• Identifying and Graphing Cardioids
• Difference Quotient
• Finding Upper and Lower Bounds
• Evaluating Functions
• Right Triangle Trigonometry
• Arithmetic Sequences/Progressions
• Geometric Sequences/Progressions
• Finding the Next Term of the Sequence
• Finding the nth Term Given a List of Numbers
• Finding the nth Term
• Finding the Sum of First n Terms
• Expanding Series Notation
• Finding the Sum of the Series
• Finding the Sum of the Infinite Geometric Series
• Converting to Rectangular Coordinates
• Evaluating Limits Approaching a Value
• Evaluating Limits Approaching Infinity
• Finding the Angle Between the Vectors
• Determining if the Point is on the Graph
• Finding the Antiderivative
• Checking if Continuous Over an Interval
• Determining if a Series is Divergent
• Using the Integral Test for Convergence
• Determining if an Infinite Series is Convergent Using Cauchy's Root Test
• Using the Limit Definition to Find the Tangent Line at a Given Point
• Finding the nth Derivative
• Finding the Derivative Using Product Rule
• Finding the Derivative Using Quotient Rule
• Finding the Derivative Using Chain Rule
• Use Logarithmic Differentiation to Find the Derivative
• Finding the Derivative
• Implicit Differentiation
• Using the Limit Definition to Find the Derivative
• Evaluating the Derivative
• Finding Where dy/dx is Equal to Zero
• Finding the Linearization
• Finding a Tangent Line to a Curve
• Checking if Differentiable Over an Interval
• The Mean Value Theorem
• Finding the Inflection Points
• Find Where the Function Increases/Decreases
• Finding the Critical Points of a Function
• Find Horizontal Tangent Line
• Evaluating Limits with L'Hospital Rule
• Local Maxima and Minima
• Finding the Absolute Maximum and Minimum on the Given Interval
• Finding Concavity using the Second Derivative
• Finding the Derivative using the Fundamental Theorem of Calculus
• Find the Turning Points
• Finding the Integral
• Evaluating Definite Integrals
• Evaluating Indefinite Integrals
• Substitution Rule
• Finding the Arc Length
• Finding the Average Value of the Derivative
• Finding the Average Value of the Equation
• Finding Area Between Curves
• Finding the Volume
• Finding the Average Value of the Function
• Finding the Root Mean Square
• Integration by Parts
• Trigonometric Integrals
• Trigonometric Substitution
• Integration by Partial Fractions
• Eliminating the Parameter from the Function
• Verify the Solution of a Differential Equation
• Solve for a Constant Given an Initial Condition
• Find an Exact Solution to the Differential Equation
• Verify the Existence and Uniqueness of Solutions for the Differential Equation
• Solve for a Constant in a Given Solution
• Solve the Bernoulli Differential Equation
• Solve the Linear Differential Equation
• Solve the Homogeneous Differential Equation
• Solve the Exact Differential Equation
• Approximate a Differential Equation Using Euler's Method
• Finding Elasticity of Demand
• Finding the Consumer Surplus
• Finding the Producer Surplus
• Finding the Gini Index
• Finding the Geometric Mean
• Finding the Quadratic Mean (RMS)
• Find the Mean Absolute Deviation
• Finding the Mid-Range (Mid-Extreme)
• Finding the Interquartile Range (H-Spread)
• Finding the Midhinge
• Finding the Standard Deviation
• Finding the Skew of a Data Set
• Finding the Range of a Data Set
• Finding the Variance of a Data Set
• Finding the Class Width
• Solving Combinations
• Solving Permutations
• Finding the Probability of Both Independent Events
• Finding the Probability of Both Dependent Events
• Finding the Probability for Both Mutually Exclusive Events
• Finding the Conditional Probability for Independent Events
• Determining if Given Events are Independent/Dependent Events
• Determining if Given Events are Mutually Exclusive Events
• Finding the Probability of Both not Mutually Exclusive Events
• Finding the Conditional Probability Using Bayes' Theorem
• Finding the Probability of the Complement
• Describing Distribution's Two Properties
• Finding the Expectation
• Finding the Variance
• Finding the Probability of a Binomial Distribution
• Finding the Probability of the Binomial Event
• Finding the Mean
• Finding the Relative Frequency
• Finding the Percentage Frequency
• Finding the Upper and Lower Class Limits of the Frequency Table
• Finding the Class Boundaries of the Frequency Table
• Finding the Class Width of the Frequency Table
• Finding the Midpoints of the Frequency Table
• Finding the Mean of the Frequency Table
• Finding the Variance of the Frequency Table
• Finding the Standard Deviation of the Frequency Table
• Finding the Cumulative Frequency of the Frequency Table
• Finding the Relative Frequency of the Frequency Table
• Finding the Median Class Interval of the Frequency Table
• Finding the Modal Class of the Frequency Table
• Creating a Grouped Frequency Distribution Table
• Finding the Data Range
• Finding a z-Score for a Normal Distribution
• Approximating Using Normal Distribution
• Finding the Probability of the z-Score Range
• Finding the Probability of a Range in a Nonstandard Normal Distribution
• Finding the z-Score Using the Table
• Finding the z-Score
• Testing the Claim
• Finding a t-Value for a Confidence Level
• Finding the Critical t-Value
• Setting the Alternative Hypothesis
• Setting the Null Hypothesis
• Determining if Left, Right, or Two Tailed Test Given the Null Hypothesis
• Determining if Left, Right, or Two Tailed Test Given the Alternative Hypothesis
• Finding Standard Error
• Finding the Linear Correlation Coefficient
• Determining if the Correlation is Significant
• Finding a Regression Line
• Cramer's Rule
• Solving using Matrices by Elimination
• Solving using Matrices by Row Operations
• Solving using an Augmented Matrix
• Finding the Simple Interest Received
• Finding the Present Value with Compound Interest
• Finding the Simple Interest Future Value
• Finding the Future Value with Continuous Interest
• Finding the Norm in Real Vector Space
• Finding the Direction Angle of the Vector
• Finding the Cross Product of Vectors
• Finding the Dot Product of Vectors
• Determining if Vectors are Orthogonal
• Finding the Distance Between the Vectors
• Finding a Unit Vector in the Same Direction as the Given Vector
• Finding the Angle Between Two Vectors Using the Cross Product
• Finding the Angle Between Two Vectors Using the Dot Product
• Finding the Projection of One Vector Onto another Vector
• Matrices Addition
• Matrices Subtraction
• Matrices Multiplication
• Finding the Trace
• Finding the Basis
• Matrix Dimension
• Convert to a Linear System
• Diagonalizing a Matrix
• Determining the value of k for which the system has no solutions
• Linear Independence of Real Vector Spaces
• Finding the Null Space
• Determining if the Vector is in the Span of the Set
• Finding the Number of Protons
• Finding the Number of Electrons
• Finding the Number of Neutrons
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How to Solve Math Problems

Last Updated: May 16, 2023 Fact Checked

This article was co-authored by Daron Cam . Daron Cam is an Academic Tutor and the Founder of Bay Area Tutors, Inc., a San Francisco Bay Area-based tutoring service that provides tutoring in mathematics, science, and overall academic confidence building. Daron has over eight years of teaching math in classrooms and over nine years of one-on-one tutoring experience. He teaches all levels of math including calculus, pre-algebra, algebra I, geometry, and SAT/ACT math prep. Daron holds a BA from the University of California, Berkeley and a math teaching credential from St. Mary's College. This article has been fact-checked, ensuring the accuracy of any cited facts and confirming the authority of its sources. This article has been viewed 573,118 times.

Although math problems may be solved in different ways, there is a general method of visualizing, approaching and solving math problems that may help you to solve even the most difficult problem. Using these strategies can also help you to improve your math skills overall. Keep reading to learn about some of these math problem solving strategies.

Understanding the Problem

• Draw a Venn diagram. A Venn diagram shows the relationships among the numbers in your problem. Venn diagrams can be especially helpful with word problems.
• Draw a graph or chart.
• Arrange the components of the problem on a line.
• Draw simple shapes to represent more complex features of the problem.

Developing a Plan

Solving the Problem

Expert Q&A

• Seek help from your teacher or a math tutor if you get stuck or if you have tried multiple strategies without success. Your teacher or a math tutor may be able to easily identify what is wrong and help you to understand how to correct it. Thanks Helpful 1 Not Helpful 1
• Keep practicing sums and diagrams. Go through the concept your class notes regularly. Write down your understanding of the methods and utilize it. Thanks Helpful 1 Not Helpful 0

You Might Also Like

• ↑ Daron Cam. Math Tutor. Expert Interview. 29 May 2020.
• ↑ http://www.interventioncentral.org/academic-interventions/math/math-problem-solving-combining-cognitive-metacognitive-strategies
• ↑ http://tutorial.math.lamar.edu/Extras/StudyMath/ProblemSolving.aspx
• ↑ https://math.berkeley.edu/~gmelvin/polya.pdf

About This Article

To solve a math problem, try rewriting the problem in your own words so it's easier to solve. You can also make a drawing of the problem to help you figure out what it's asking you to do. If you're still completely stuck, try solving a different problem that's similar but easier and then use the same steps to solve the harder problem. Even if you can't figure out how to solve it, try to make an educated guess instead of leaving the question blank. To learn how to come up with a solid plan to use to help you solve a math problem, scroll down! Did this summary help you? Yes No

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How to Ace Math Problem Solving

When your kids struggle with their math, it’s time to take a step back and take a deep breath. They need to slow down and take their time. Here’s a step by step guide that will help your kids get through those tough math problems.

We’ll use a grade 3 addition word problem as an example to clarify:

Pinky the Pig bought 36 apples while Danny the Duck bought 73 apples and 14 bananas. How many apples do they have altogether?

Read the problem

Carefully read through the problem to make sure you understand what is being asked.

Pinky the pig and Danny the duck bought apples and bananas. The question is how many apples they have together.

Re-read the problem

Read through the problem again and as you read through it, make notes.

Pinky the pig –36 apples. Danny the duck –73 apples and 14 bananas. How many apples together?

What is the problem asking

In your own words, say or write down exactly what the question is asking you to solve.

The question is asking how many apples the pig and the duck bought together.

Write it down in detail

Go through the problem and write out the information in an organized fashion. A diagram or table might help.

Turn it into math

Figure out what math operation(s) or formula(s) you need to use in order to solve this problem.

The problem wants us to add the number of apples Pinky the Pig and Danny the Duck have together. That means we need to make use of addition to add the apples.

Find an example

Are you still struggling? Sometimes it’s hard to work out the solution, especially if the math problem involves several steps. It’s time to present the problem in an easier way. As teachers and parents we can often help our kids simplify the problem from our own math knowledge. If the problem is a bit harder, there are lots of resources online that you can look up for similar problems that have been worked out on paper or a video tutorial to watch.

In our example, let’s say the double-digit numbers are intimidating our student, so we’re going to simplify the equation for the sake of helping our student understand the operation needed.

Let’s say Pinky the Pig bought 3 apples and Danny the Duck 7 apples and 1 banana. Now, how many apples have they bought together? With 3 apples and 7 apples bought, the total number of apples is 10.

Work out the problem

Now that we have got to the bottom of what is being asked and know what operation to use, it’s time to work out the problem.

Pinky the Pig bought 36 apples. Danny the Duck bought 73 apples. (The 14 bananas do not matter) We need to add up the apples. 36 + 73 = 109

Check and review your answer

Check that your answer is correct. Always ask: does this answer make sense?  You can use estimation using mental math, for example.

Let’s round the numbers: 30 + 70 = 100. That is close to the exact number so it’s in the correct range.

The beauty of the basic operations is that addition and subtraction can be used to check answers too.

If we use the sum and take away one of the numbers, it should equal the other number.

109 – 73 = 36 109 – 36 = 73

If our student did not work out the sum correctly, we would not come to these sums.

(By the way, the same can be done with multiplication and division.)

Finally, go back and review the problem one last time. By going over the concepts, operations and formulas, it will help your kids to internalize the process and help them tackle harder math problems in the future.

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Google can now solve trickier math problems for you with these new features

Math is a challenging subject because it requires an understanding of how to perform the operation to reach an answer, which makes it more difficult to Google an equation to find the answer difficult -- until now. 

Google added new updates to Search and Lens that make it easier for users to get assistance when solving math problems. All users have to do now is type the equation or integral into the Search bar, or take a picture with Lens to get a step-by-step explanation or solution. 

Also:  Chrome on iOS unveils a much-anticipated feature. Here's how to access it

To test out the experience for yourself, on desktop, you can type in an equation or type the term "Math Solver" on Google Search where you will be prompted to enter a math problem or select from the examples to see how it works. The math solver experience will be coming to mobile soon. 

Lens can also be leveraged by users to take a photo of geometry triangle problems, solving the challenge of trying to put primarily visual problems into words. 

Advancements in Google's large language models also give Search the capability to solve word problems. 

All you have to do is type the problem into Search, where you will be met with steps that tell you how to solve the problem by identifying the known and unknown values and providing correct formulas. 

Also: The AI I want to see in the world: 5 ways it could manage my Gmail inbox for me

Lastly, Google is also making it easier to explore STEM-related concepts on Search by including 3D models and interactive diagrams for almost 1,000 biology, chemistry, physics, astronomy, and related topics, according to Google. 

For example, if you Google "mitochondrion" you will have the opportunity to click on and learn from an interactive diagram that provides an overview, as well as specific details about the individual parts. 

Google's new tools help users verify the authenticity of images online faster

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How Google's AI Bard helped me fix a Gmail technical problem

Google Search can now help you solve geometry, physics and calculus problems

Google updated its search engine and Lens tool with new features to help you visualize and solve problems in more difficult subjects like geometry, physics, trigonometry and calculus.

The update allows you to type an equation into the Search bar or take a photo with Lens to get the correct answer and a step-by-step explanation. The new feature also provides solutions to word problems. For instance, the solver can explain high school physics concepts like finding the average acceleration using the equation “vf= vi + at.”

You can also access the math solver feature by typing “[math solver]” directly into the search bar. However, this function is only available on desktop. It will launch on mobile by the end of the year, the company told TechCrunch.

Another useful new feature is the geometry solver. Since solving geometry problems requires a lot of visualization, Lens can now explain both the visual and the text components of the problem.

However, Lens can only interpret “certain triangle problems,” the company wrote in its blog post, attributed to Google product lead Robert Wong.

We reached out to Wong to clarify, who told us that the geometry solver can currently do the following:

• Find the area of a triangle using A = 1/2 x height x base or Heron’s formula.
• Find the perimeter of a triangle.
• Apply the “Isosceles Triangle Theorem” to determine an unknown side length or angle.
• Apply the “Pythagorean Theorem” to figure out the side lengths in right triangles.

“For our geometry experience, we wanted to focus on a subset of problems that could have a large impact before eventually expanding to cover a more diverse range of geometry problems,” Wong added.

Image Credits: Google

For the physics and geometry solver, Google leveraged its Multitask Unified Model (MUM), which understands specific search queries across various formats, including text, images and videos.

“Language and query understanding are complex challenges that require sophisticated systems to ensure we’re producing the most helpful results — especially in cases where our systems are looking at information in multiple different formats,” Wong explained. “In this case, these features need to understand information across both text and images to correctly interpret a problem and provide accurate step-by-step responses that include both text and diagrams.”

When asked about the accuracy rate for the new experience, the company declined to share specific numbers.

“Our testing shows the accuracy rate is very high,” Wong said.

In addition to problem-solving, Google also launched new 3D models on Search, letting you examine interactive diagrams related to nearly 1,000 topics, such as biology, chemistry, physics, astronomy and more. Google introduced 3D models in 2021, rolling out diagrams for more than 200 chemistry, biology, physics and anatomy concepts.

Over the years, the tech giant has launched multiple features that help you learn, including its interactive feature, “practice problems,” that tests your knowledge of high school topics like math, chemistry and physics. Google Search recently rolled out an English tutoring tool that provides prompts and asks you to say the answers out loud using a provided vocabulary word.

Google takes aim at Duolingo with new English tutoring tool

Android Police

Google's ai can now solve word problems you paste into the search bar.

Lens and Search just got a major math makeover

• Google's new features in Search and Lens make it easier for users to visualize STEM concepts and solve complex equations.
• Search can now handle physics word problems, and Lens can interpret visual problems in geometry, providing step-by-step guides on how to solve them.
• Google's AI capabilities, powered by technology like PaLM, contribute to the development of these new tools, but they should be used as supplementary resources alongside traditional learning methods.

Both students and working professionals frequently rely on search engines to find solutions to intricate queries. Google, in its efforts to maintain its edge as the world's leading search engine, has recently upgraded its features to better serve those in STEM fields. As of recently, if you're grappling with a challenging math or physics problem, Google's Search and Lens might just be your new best friends.

Google has introduced new features in both Search and Lens to assist users in visualizing STEM-related concepts and determining the right equations for their problems. Whether you're trying to decipher a complicated physics concept or a perplexing geometry problem, these tools aim to make the process more intuitive.

For instance, if you're stuck on a calculus problem, you can now simply type your equation into the Search bar or snap a photo with Lens to receive a step-by-step explanation and solution. This feature is not limited to just equations; it extends to word problems , especially those from high-school physics topics.

The integration of Lens, in particular, addresses a unique challenge in geometry. Describing visual problems using words can be cumbersome. For example, if you're given a diagram of a triangle with measurements of two sides and need to find its area, Lens can now interpret both the visual and text components of the problem, offering a comprehensive step-by-step guide on how to solve it.

This advancement isn't just about problem-solving. Google has also introduced new interactive 3D models on Search, allowing users to visually explore almost 1,000 topics from biology, chemistry, physics, and astronomy — an example given by the company was a search for the term "mitochondria." This feature can be particularly useful for those who wish to gain a deeper visual understanding of complex STEM concepts.

The underlying technology that powers these features can be traced back to Google's efforts in enhancing its AI capabilities. A notable mention on this front is the integration of PaLM into Bard . Initially, Bard, based on Google’s Language Model for Dialogue Application (LaMDA), was more adept at holding conversations than logical reasoning. However, with the fusion of PaLM into its code base, Bard's capabilities expanded to include arithmetic, code completion, semantic parsing, logical inference, and more. We could be seeing a similar implementation of machine learning here with the new tools in Search and Lens.

While AI tools like the features Google is introducing today with Search and Lens are becoming increasingly sophisticated, it's essential for users to approach them as supplementary resources, complementing traditional learning and problem-solving methods. As technology continues to evolve, the line between human and machine capabilities might blur, but the essence of learning and understanding will always remain inherently human.

Easy Finger Math Tricks to Help Kids Solve Problems

Posted: November 9, 2023 | Last updated: November 9, 2023

While using your fingers isn't the fastest way to recall a multiplication fact while doing a problem, finger math tricks can help kids figure out how to answer the problem at hand — and as they work on their math, they will eventually learn all the facts by repetition.

Note that before your child can understand other finger tricks, they must be able to count by 2s, 5s, and 10s and multiply by 2s, 3s, and 4s.

Quick Finger Math Tricks for Threes and Fours

The tricks for multiplying by threes and fours are really a matter of counting out the answer on your fingers. As your children count out the answer repeatedly, they'll memorize it and then be able to move on to larger numbers.

Multiplying by Three

Did you realize that all of your fingers have three segments? Therefore, you can figure out anything from 3 x 1 to 3 x 10 by counting the segments on each finger. To start:

• Hold up the number of fingers you're going to multiply by 3. For example, if the problem is 3 x 4 — hold up four fingers.
• Count each segment on each finger you're holding up, and you should come up with 12 — which is the correct answer.

Multiplying by Four

Multiplying by four is the same as multiplying by two — twice. To start:

• Hold up the number of fingers to correspond with the number you are multiplying by four. For example, if you are multiplying 4 x 6 — hold up six fingers.
• Count each finger by two, moving from left to right. Then count each finger again, continuing to count by twos, until you've counted every finger twice.

Helpful Hack To keep track of the fingers you've counted twice, sometimes it's easier to put your finger down as you count the first time, and back up as you count the second time.

Finger Math Tricks for Multiplying by 6, 7, 8, and 9

While numbers one through five are easy for most kids to remember, six and up often pose a problem. This handy trick will make it a little easier to work those problems out.

Multiplying 6, 7, 8, and 9 by Hand

To begin, assign each finger a number. For example, your thumbs represent 6, your index fingers each represent 7, etc. This will remain the same throughout the finger math hack.

Your left hand will represent the first number that you are multiplying and your right hand will represent the second number you are multiplying. In this example, we are multiplying 7 x 8. 

To Determine the Part of Your Answer:

• On your left hand, put down the finger that represents the number you are multiplying as well as any fingers whose number value is less than this figure. In this example, you are multiplying 7 x 8, so the left hand will represent 7. You will drop your index finger (number 7) and your thumb (number 6).
• Similarly, the right hand will represent eight, so you will drop down your middle finger (number 8), your index finger (number 7), and your thumb (number 6).
• Now, just multiply the fingers that are still pointed upwards. In this case, you will have three fingers on your left hand and two on your right, so you will multiply 3 x 2 to get 6. This is the first part of your answer!

To Determine the Second Part of Your Answer:

• Keeping your fingers in the same positions, count how many fingers are folded down. In the 7 x 8 example, you should have five fingers folded. 
• You will count each of these in quantities of ten. So, 10, 20, 30, 40, 50.
• 50 is your answer.

To Determine Your Final Answer:

• Add your two numbers together. In this example, you would add 6 + 50, which gives you 56!

Another Finger Math Trick Just for Nine

There is a trick that works separately, just for multiplying by the number nine.

• To start, hold up all ten fingers, with your palms facing you.
• Assign each finger a number, starting with your left-hand thumb and ending with your right-hand thumb. The left-hand thumb will be one, the left-hand index finger will be two, and so on until you reach the number 10 for your right-hand thumb.
• To tackle a problem, put down the corresponding finger of the number you're multiplying by nine. For example, if you are multiplying 9 x 8, you'd put down the eighth finger (which will be on your right hand).
• Count all the fingers to the left of the finger you have folded down. This will give you 7. This is the first digit of your answer.
• Count all the fingers to the right of the finger you have folded down. This will give you 2. This is the second digit of your answer.
• Put the numbers together! Your answer is 72.

Finger Multiplication Tricks Can Make Math Easy and Fun

While the hope is that your kids will eventually memorize their multiplication charts , using some quick hand tricks for multiplication and letting them count things out on their fingers is not a bad way to learn. It keeps frustration at bay since the answer is always a fingertip away, and the repetition of having to figure it out will help cement those facts into their brains.

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