Algebra Calculator

Q: What is the quadratic formula?

The quadratic formula solves any equation shaped like ax² + bx + c = 0 : x = (−b ± √(b² − 4ac)) / (2a) Put in your values for a , b , and c . The ± sign gives you two answers. For 2x² + 3x − 2 = 0 , you get x = 1/2 and x = −2.

Q: What is the discriminant and what does it tell you?

The discriminant is the part under the square root: b² − 4ac . It tells you how many real answers a quadratic has. Positive: two different real answers Zero: one real answer (a repeated root) Negative: no real answers, only two complex ones

Q: Why do you flip the sign when you divide an inequality by a negative number?

Because multiplying or dividing by a negative flips the order of numbers. Look at 2 < 5. Multiply both sides by −1 and you get −2 and −5. Now −2 is bigger, so the true statement is −2 > −5. Flipping the sign keeps the statement true. You do not flip when you add or subtract.

Q: How do you complete the square?

Take x² + 6x + 4 = 0 . Move the 4 over: x² + 6x = −4 . Cut the middle number in half (6 ÷ 2 = 3), square it (9), and add 9 to both sides. Now the left side is a perfect square: (x + 3)² = 5 . Take the square root of both sides and solve, giving x = −3 ± √5.

Q: What is the difference between long division and synthetic division of polynomials?

Long division works for any divisor, like x² + 1 or 2x − 5 . Synthetic division is a shortcut that only works when you divide by x − c , a linear factor with a 1 in front. Synthetic division uses just the coefficients, so it is faster, but long division is the safe choice for anything else.

Q: How do you find the vertex of a parabola?

For y = ax² + bx + c , the vertex x-value is x = −b / (2a) . Put that x back in to get the y-value. For x² − 4x + 1 , x = 4/2 = 2 and y = 4 − 8 + 1 = −3, so the vertex is (2, −3). If a is positive the vertex is the lowest point; if a is negative it is the highest point.

Q: How do you find the slope from an equation like 3x + 2y = 12?

Solve for y so the equation looks like y = mx + b . Here, 2y = −3x + 12 , so y = −3/2 x + 6 . The number in front of x is the slope, so m = −3/2. The number alone, 6, is the y-intercept.

Algebra Calculator

Updated Sep 4, 2026 By Jehan Wadia

Introduction

This free online Algebra Calculator solves math problems and shows you the steps. Type an equation like x^2 - 5x + 6 = 0, pick what you want done, and the answer appears with the working.

It works with many kinds of algebra problems:

Equations: linear, quadratic, polynomial, rational, and absolute value
Inequalities: with answers written as intervals
Systems of equations: by substitution, elimination, Cramer's Rule, or Gaussian elimination
Basic operations: simplify, factor, and expand
Polynomials: add, multiply, long division, and synthetic division
Rational expressions: combine fractions and split them with partial fractions
Functions and graphs: roots, intercepts, slope, vertex, range, inverse, tangent lines, and plots

You can also switch answers between fractions and decimals, move sliders to see how numbers change a result, check your own answer, and browse a topic list of ready-made examples. If your problem is on paper, use the photo input to snap a picture and type the expression in.

How to use our Algebra Calculator

Type a math problem, pick what you want the calculator to do, and press Calculate. You get the answer, the step-by-step work, and a graph when one fits the problem.

Expression or equation box: Type your problem here, like x^2 - 5x + 6 = 0, 2x + 3 > 7, or (x + 3)(x - 4). Use ^ for powers, * for times, / for divide, and | | for absolute value.

Mode buttons: Choose the job you want done, such as Solve, Simplify, Factor, Expand, Roots, Divide, Graph, or Evaluate. The steps you see match the mode you pick.

Topics list: Open the topic tree and click a lesson, like Quadratic Formula or Partial Fractions. It fills the box with a ready example and sets the right mode for you.

Solve for variable: When a problem has more than one letter, pick the letter you want to solve for, like solving A = P(1 + r) for r.

Variable values: In Evaluate mode, type a number for each letter, such as a = 3 and n = 10. The calculator plugs them in and shows the value.

Tangent x value: In Tangent mode, type the x where you want the tangent line. The tool gives the slope and the line at that point.

Graph window (Xmin, Xmax, Ymin, Ymax): Set how far the graph goes left, right, down, and up. Click Redraw to see the new view.

Sliders switch: Turn sliders on to drag values for each letter. The live line shows the value, or tells you if the statement is true or false.

Fraction or decimal toggle: Click it to switch answers between exact fractions and decimals.

Check an answer box: Type a value, like x = 3, to test it in your equation. The tool tells you if it works.

Image input: Upload or snap a photo, drag a box around the problem, then fix the text and load it into the calculator.

Calculate and Reset: Press Calculate to solve, or Reset to clear everything and start over.

What Is Algebra?

Algebra is the part of math that uses letters to stand for numbers. A letter like x is called a variable. It holds the spot of a number you do not know yet. Your job is to find out what that number is.

Expressions, Equations, and Inequalities

Expression: a mix of numbers, letters, and signs, like 3x + 5. It has no equals sign, so you can only simplify it.
Equation: two expressions set equal, like 3x + 5 = 11. You solve it to find the value of the variable.
Inequality: uses <, >, ≤, or ≥, like 2x + 3 > 7. The answer is a range of numbers, not just one.

How to Solve an Equation

Think of an equation as a balance scale. Whatever you do to one side, you must do to the other side. Add, subtract, multiply, or divide both sides until the variable is alone. Then check your answer by putting it back into the first equation.

Main Algebra Topics

Linear equations: the variable has a power of 1, like 4x + 2 = 2(x + 6). Their graphs are straight lines.
Quadratic equations: they have an x² term, like x² − 5x + 6 = 0. You can solve them by factoring, by completing the square, or with the quadratic formula. Their graphs are U-shaped curves called parabolas.
Polynomials: sums of terms with whole-number powers, like 5x⁴ − 2x² + 7. You can add, subtract, multiply, and divide them.
Factoring: writing an expression as a product. For example, x² − 9 becomes (x + 3)(x − 3). Factoring makes hard problems easier.
Rational expressions: fractions with variables, like 1/x + 1/(x + 2). The bottom can never equal zero.
Systems of equations: two or more equations at once. You can solve them by substitution, elimination, Cramer's rule, or Gaussian elimination.
Absolute value: the distance from zero, so |2x − 3| = 7 gives two cases and often two answers.
Sequences: number patterns. Arithmetic ones add the same amount each time; geometric ones multiply by the same amount.

Functions and Graphs

A function takes an input and gives one output. When you graph a function, you can see key parts: the x-intercepts (where it crosses the x-axis, also called roots or zeroes), the y-intercept, the slope of a line, and the vertex (the top or bottom point of a parabola). The domain is the set of inputs that work, and the range is the set of outputs you get.

Order of Operations

Always work in this order: parentheses, exponents, multiply and divide (left to right), then add and subtract (left to right). Skipping this order is the most common way to get a wrong answer.

Why Algebra Matters

Algebra is the base for geometry, trigonometry, statistics, and calculus. People use it every day to work out money, loans, distances, speeds, recipes, and building plans. Once you can solve for a missing number, you can solve real problems.

Formulas used

Quadratic formula (solving ax^2 + bx + c = 0)

x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

Vertex of a quadratic

x_v = -\frac{b}{2a}, \qquad y_v = c - \frac{b^2}{4a}

Slope from a linear equation Ax + By = C

m = -\frac{A}{B}, \qquad y = mx + b

Tangent line at x = x_0

y = f(x_0) + f'(x_0)\,(x - x_0)

Cramer's rule for a 2x2 system

x = \frac{c_1 b_2 - c_2 b_1}{a_1 b_2 - a_2 b_1}, \qquad y = \frac{a_1 c_2 - a_2 c_1}{a_1 b_2 - a_2 b_1}

Arithmetic sequence: n-th term and sum

a_n = a + (n - 1)d, \qquad S_n = \frac{n}{2}\left(2a + (n - 1)d\right)

Geometric sequence: n-th term

a_n = a \cdot r^{\,n - 1}

Live check of an equality (relative tolerance test)

\frac{\left|L - R\right|}{\max\left(\left|L\right|, \left|R\right|, 1\right)} < 10^{-9}

Frequently asked questions

What is the quadratic formula?

The quadratic formula solves any equation shaped like ax² + bx + c = 0:

x = (−b ± √(b² − 4ac)) / (2a)

Put in your values for a, b, and c. The ± sign gives you two answers. For 2x² + 3x − 2 = 0, you get x = 1/2 and x = −2.

What is the discriminant and what does it tell you?

The discriminant is the part under the square root: b² − 4ac. It tells you how many real answers a quadratic has.

Positive: two different real answers
Zero: one real answer (a repeated root)
Negative: no real answers, only two complex ones

Why do you flip the sign when you divide an inequality by a negative number?

Because multiplying or dividing by a negative flips the order of numbers. Look at 2 < 5. Multiply both sides by −1 and you get −2 and −5. Now −2 is bigger, so the true statement is −2 > −5. Flipping the sign keeps the statement true. You do not flip when you add or subtract.

How do you factor a trinomial like x² + 5x + 6?

Find two numbers that multiply to the last term and add to the middle term. For x² + 5x + 6, the numbers 2 and 3 multiply to 6 and add to 5. So it factors as (x + 2)(x + 3).

When the front number is not 1, like 6x² + 11x − 35, multiply a times c (6 × −35 = −210), find two numbers that multiply to that and add to 11, then factor by grouping.

How do you complete the square?

Take x² + 6x + 4 = 0. Move the 4 over: x² + 6x = −4. Cut the middle number in half (6 ÷ 2 = 3), square it (9), and add 9 to both sides. Now the left side is a perfect square: (x + 3)² = 5. Take the square root of both sides and solve, giving x = −3 ± √5.

What is an extraneous solution?

It is an answer that shows up in your work but does not really work in the first equation. This happens a lot with rational equations and square roots. For example, if you solve a fraction equation and get x = −2, but the bottom of the fraction is x + 2, then x = −2 makes it zero. That answer must be thrown out. Always check answers in the original equation.

What is the difference between long division and synthetic division of polynomials?

Long division works for any divisor, like x² + 1 or 2x − 5. Synthetic division is a shortcut that only works when you divide by x − c, a linear factor with a 1 in front. Synthetic division uses just the coefficients, so it is faster, but long division is the safe choice for anything else.

How do you find the vertex of a parabola?

For y = ax² + bx + c, the vertex x-value is x = −b / (2a). Put that x back in to get the y-value. For x² − 4x + 1, x = 4/2 = 2 and y = 4 − 8 + 1 = −3, so the vertex is (2, −3). If a is positive the vertex is the lowest point; if a is negative it is the highest point.

How do you find the slope from an equation like 3x + 2y = 12?

Solve for y so the equation looks like y = mx + b. Here, 2y = −3x + 12, so y = −3/2 x + 6. The number in front of x is the slope, so m = −3/2. The number alone, 6, is the y-intercept.

How do you solve an absolute value equation?

Split it into two cases, because the inside can be positive or negative. For |2x − 3| = 7, write 2x − 3 = 7 and 2x − 3 = −7. Solving gives x = 5 and x = −2. If the right side is negative, like |x| = −4, there is no solution, since distance is never negative.

How do you solve an absolute value inequality?

Less-than problems become one range in the middle. |x + 1| ≤ 4 means −4 ≤ x + 1 ≤ 4, so −5 ≤ x ≤ 3.

Greater-than problems break into two separate pieces. |x| > 3 means x < −3 or x > 3.

What is interval notation?

It is a short way to write a range of numbers. Round brackets mean the endpoint is not included, and square brackets mean it is. So x > 2 is written (2, ∞), and −2 ≤ x ≤ 2 is written [−2, 2]. Use ∪ to join two separate pieces, like (−∞, −3) ∪ (3, ∞).

How many solutions can a system of two equations have?

Three possibilities. If the lines cross once, there is one solution. If the lines are parallel, there is no solution. If both equations describe the same line, there are infinitely many solutions. When you solve and get a false statement like 0 = 5, there is no solution. If you get 0 = 0, there are infinite solutions.

What are partial fractions used for?

Partial fractions split one messy fraction into simpler ones added together. For example, 1/(x² − 1) becomes 1/2 · 1/(x − 1) − 1/2 · 1/(x + 1). This makes the fraction much easier to work with, and it is a key step in calculus integration.

How do you find the inverse of a function?

Swap x and y, then solve for y. Start with y = (x − 4)/3. Swap: x = (y − 4)/3. Multiply by 3: 3x = y − 4. Add 4: y = 3x + 4. The inverse undoes the original function, so plugging one into the other gives you back x.

What is the domain of a rational expression?

The domain is every number you may put in for x. For a fraction with variables, you must leave out any value that makes the bottom zero, because you cannot divide by zero. In 1/(x + 2), x cannot be −2. So the domain is all real numbers except −2.

Are roots, zeros, and x-intercepts the same thing?

Yes, they all point to the same idea: the x-values that make the expression equal zero. "Roots" and "zeros" are used for equations and functions, while "x-intercepts" is used for graphs, where the curve crosses the x-axis. For x² − 5x + 6, the roots are 2 and 3, and the graph crosses at (2, 0) and (3, 0).

What does FOIL mean in algebra?

FOIL is a memory trick for multiplying two binomials: First, Outer, Inner, Last. For (x + 3)(x − 4): First x·x = x², Outer x·(−4) = −4x, Inner 3·x = 3x, Last 3·(−4) = −12. Add them up and combine: x² − x − 12.

What are the main exponent rules?

These four cover most problems:

Multiply: x^a · x^b = x^a+b
Divide: x^a / x^b = x^a−b
Power of a power: (x^a)^b = x^ab
Zero power: x⁰ = 1 (when x is not 0)

So (x³ · x⁴)/x² = x⁷/x² = x⁵.

How do you solve for one variable in a formula with several letters?

Treat the other letters like plain numbers and undo the operations step by step. To solve A = P(1 + r) for r: divide both sides by P to get A/P = 1 + r, then subtract 1 to get r = A/P − 1. The goal is the same as any equation, get your letter alone on one side.

What is the formula for the nth term of an arithmetic sequence?

Use a_n = a + (n − 1)d, where a is the first term, d is the amount you add each time, and n is the term number. With a = 3, d = 5, and n = 10, you get 3 + 9(5) = 48. For a geometric sequence you multiply instead: a_n = a · rⁿ⁻¹.

What is the difference between factoring and expanding?

They are opposites. Expanding multiplies things out: (x + 3)(x − 4) becomes x² − x − 12. Factoring goes backward and writes a sum as a product: x² − x − 12 becomes (x + 3)(x − 4). Expanding helps you combine like terms; factoring helps you find roots.

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