Algebra Solver

Choose a solver mode, enter your equation, and get instant step-by-step solutions with verification

🧮 Solver Mode

📋 Quick Examples

2x + 5 = 13

3x - 7 = 2x + 4

5(x+3) = 2x+21

x/3 + 4 = 7

4x+2 = 3x-5

Enter Linear Equation (solve for x)

Use x as the variable. Supports: 2x+5=13 · 3(x-2)=9 · x/4+1=3 · 2x-3=x+7

Invalid input.

✅ SOLUTION

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Exact Solution

Solution Summary

Step-by-Step Solution

Verification — Substitute Back

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What Is Algebra? Types of Equations Explained

A complete guide to linear, quadratic, and simultaneous equations — with real-world examples and solving strategies

Overview

Algebra: The Language of Unknowns

Algebra is the branch of mathematics dealing with symbols (variables) and the rules for manipulating those symbols. At its core, algebra allows us to express and solve problems involving unknown quantities. The variable x (or any letter) acts as a placeholder for a number we want to find.

Every equation in algebra states that two expressions are equal. Solving an equation means finding the value(s) of the variable that make the equality true. The strategies differ based on the degree of the equation — the highest power of the variable:

📐 Degree matters: A linear equation (degree 1, like 2x+5=13) has exactly one solution. A quadratic (degree 2, like x²-5x+6=0) has up to two solutions. A cubic (degree 3) has up to three. The degree determines which method to use and how many solutions to expect.

The golden rule of algebra: whatever you do to one side of an equation, you must do to the other. This preserves equality. Add, subtract, multiply, or divide both sides by the same value, and the equation remains balanced — leading you step by step to the solution.

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Linear (Degree 1)

Form: ax + b = c. One variable, one solution. Solved by isolating x through addition, subtraction, multiplication, and division. Example: 2x+5=13 → x=4.

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Quadratic (Degree 2)

Form: ax²+bx+c=0. Up to two real roots. Solved by factoring, completing the square, or the quadratic formula. Discriminant Δ=b²-4ac determines root type.

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Systems of Equations

Two or more equations with two or more unknowns. Solved by substitution, elimination, or matrix methods. Represents the intersection of lines on a graph.

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Simplify & Expand

Simplifying collects like terms into the shortest equivalent form. Expanding (using FOIL or distributive property) converts products of brackets into standard polynomial form.

Key Algebra Formulas & Identities Reference

Essential formulas every algebra student needs — from the quadratic formula to difference of squares

Formula Sheet

Must-Know Algebra Identities

Identity / Formula	Expression	Used For
Quadratic Formula	`x = (-b ± √(b²-4ac)) / 2a`	Solving ax²+bx+c=0 for any real/complex roots
Discriminant	`Δ = b² - 4ac`	Δ>0: two real roots · Δ=0: one root · Δ<0: complex
Perfect Square (sum)	`(a+b)² = a²+2ab+b²`	Expanding squared binomials
Perfect Square (diff)	`(a-b)² = a²-2ab+b²`	Expanding squared binomials
Difference of Squares	`(a+b)(a-b) = a²-b²`	Factoring & quick mental math
Sum of Cubes	`a³+b³ = (a+b)(a²-ab+b²)`	Factoring cubic polynomials
Difference of Cubes	`a³-b³ = (a-b)(a²+ab+b²)`	Factoring cubic polynomials
FOIL (binomial product)	`(a+b)(c+d) = ac+ad+bc+bd`	Expanding products of two binomials
Vieta's Formulas	`x₁+x₂ = -b/a · x₁·x₂ = c/a`	Sum and product of roots of ax²+bx+c=0
Distributive Property	`a(b+c) = ab+ac`	Expanding brackets — the foundation of algebra

How to Use the Algebra Solver

Step-by-step guide to all five solver modes and how to enter equations correctly

1
Select Your Solver Mode
Linear — equations like 2x+5=13 or 3(x-2)=9. Quadratic — equations like x²-5x+6=0. System — two equations with x and y. Simplify — collect like terms in an expression. Expand — multiply out brackets like (x+3)(x+2).
2
Use a Quick Example or Enter Your Equation
Click any quick example button to auto-fill a sample equation, or type your own. Use x^2 for x², * for multiplication, and / for division. Parentheses are supported.
3
Click "Solve — Show Step-by-Step"
The solver parses your equation, applies the correct algorithm, and displays a fully annotated step-by-step solution. Every operation is explained so you can follow the working and learn the method, not just the answer.
4
Review the Verification
After solving, the calculator automatically substitutes the answer back into the original equation to verify it is correct. This confirms accuracy and helps you check your own work.
5
Share Your Solution
Copy the full solution to your clipboard, share on WhatsApp, or tweet it. Useful for sharing solutions with classmates, tutors, or study groups.

Algebra Tips, Tricks & Common Mistakes

Powerful mental shortcuts and the most common errors students make when solving equations

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The Balance Method

Think of an equation as a balance scale. Whatever you do to one side (add, subtract, multiply, divide), do exactly the same to the other side. This is the single most important principle in all of algebra.

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Isolate the Variable

The goal is always to get x (or the variable) alone on one side. Work backwards from the equation — undo addition with subtraction, undo multiplication with division, always in reverse BODMAS order.

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Common Mistake: Sign Errors

When moving a term from one side to the other, the sign flips. Moving +5 to the other side gives -5. Moving -3x to the other side gives +3x. Forgetting this is the #1 source of algebra errors.

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Discriminant Check First

Before solving a quadratic, compute Δ = b²-4ac. If Δ<0, the equation has no real roots (complex only). If Δ=0, there's exactly one repeated root. This saves time vs. trying to factor a non-factorable quadratic.

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FOIL vs. Distributive

FOIL (First, Outer, Inner, Last) only works for two binomials. For any product of brackets, the distributive property always works: multiply each term in the first bracket by every term in the second.

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Always Verify Your Answer

Substitute your answer back into the original equation. If both sides equal the same number, your answer is correct. This step catches arithmetic mistakes and is expected in exam solutions for full marks.

Frequently Asked Questions — Algebra Solver

Answers to the most common algebra questions from students and learners

How do you solve a linear equation step by step?

To solve a linear equation like 3x + 7 = 19: (1) Subtract 7 from both sides: 3x = 12. (2) Divide both sides by 3: x = 4. (3) Verify: 3(4)+7 = 12+7 = 19 ✓. The general steps are: expand brackets, collect all x terms on one side, collect constants on the other side, then divide by the coefficient of x.

What is the quadratic formula and when should I use it?

The quadratic formula is x = (-b ± √(b²-4ac)) / 2a, used to solve ax²+bx+c=0. Use it when: (1) the quadratic doesn't factor neatly with integers, (2) the discriminant (b²-4ac) is not a perfect square, or (3) you need exact irrational or complex roots. Factoring is faster when it works, but the quadratic formula always works for any quadratic equation.

What does the discriminant tell us?

The discriminant Δ = b²-4ac of a quadratic ax²+bx+c=0 tells you the nature of the roots: if Δ>0, there are two distinct real roots (two x-intercepts on the graph); if Δ=0, there is exactly one real root (the parabola touches the x-axis); if Δ<0, there are no real roots — only complex conjugate roots (the parabola doesn't cross the x-axis).

What is the difference between simplifying and expanding?

Expanding multiplies out brackets: (x+3)(x+2) expands to x²+5x+6. Simplifying collects like terms to reduce an expression: x²+5x+6 - 3x + 1 simplifies to x²+2x+7. Expanding makes an expression longer; simplifying makes it shorter. Often you do both: first expand, then simplify the result.

How do you solve simultaneous equations?

Two main methods: (1) Elimination — multiply equations to make the coefficient of one variable equal, then add or subtract the equations to eliminate it. (2) Substitution — rearrange one equation to express one variable in terms of the other, then substitute into the second equation. This solver uses elimination for accuracy. Always verify by substituting both values into both original equations.

What does "no solution" vs "infinite solutions" mean?

No solution (inconsistent): the equations contradict each other — e.g. x+y=5 and x+y=7 cannot both be true simultaneously. On a graph, the lines are parallel. Infinite solutions (dependent): the equations are multiples of each other — e.g. x+y=5 and 2x+2y=10 are the same line. Every point on the line is a solution. One unique solution (consistent independent): the lines intersect at exactly one point — the normal case.

What is FOIL and how do I use it?

FOIL stands for First, Outer, Inner, Last — a mnemonic for multiplying two binomials. For (a+b)(c+d): First: a×c, Outer: a×d, Inner: b×c, Last: b×d. Sum these four terms. Example: (x+3)(x+2) → x², 2x, 3x, 6 → x²+5x+6. Note: FOIL only works for two binomials; for longer expressions, use the full distributive property.

Can this solver handle fractions and decimals?

Yes. For linear equations, you can enter fractions as division: x/3 + 4 = 7, or use decimal coefficients: 0.5x + 1.5 = 4. The solver handles these cases and shows working in clean fractional or decimal form as appropriate. For quadratic equations, enter integer or decimal coefficients in the standard ax²+bx+c=0 form.

Algebra Solver

Algebra Solver

Solution Summary

Step-by-Step Solution

Verification — Substitute Back

Additional Properties

Equation Details

What Is Algebra? Types of Equations Explained

Key Algebra Formulas & Identities Reference

How to Use the Algebra Solver

Select Your Solver Mode

Use a Quick Example or Enter Your Equation

Click "Solve — Show Step-by-Step"

Review the Verification

Share Your Solution

Algebra Tips, Tricks & Common Mistakes

The Balance Method

Isolate the Variable

Common Mistake: Sign Errors

Discriminant Check First

FOIL vs. Distributive

Always Verify Your Answer

Frequently Asked Questions — Algebra Solver