⚖️ Equation Solver

Solve linear and quadratic equations with step-by-step algebra. Use the tabs for ax + b = c or ax² + bx + c = 0, including real and complex roots.

Linear and Quadratic Equations — Step-by-Step Roots

BrainyCalculators editorial insight — unique to this tool

Handles ax + b = 0 and ax² + bx + c = 0 with discriminant check — negative discriminant means complex roots. Physics kinematics (s = ut + ½at²) often reduces to quadratic in time. Shows each algebraic step for homework verification.

When to use this calculator

Use when you have a full equation to solve for x. For expression simplification without solving, use Algebra.

Only solving a quadratic and need vertex details?

This page covers both linear and quadratic equations. For a dedicated quadratic formula workflow with vertex and discriminant focus, use the Quadratic Formula Calculator →

Enter coefficients for ax + b = c

x +

What is a Linear and Quadratic Equation Solver?

An equation solver finds the value or values of x that make an equation true. This page is a two-mode algebra tool: one tab solves linear equations of the form ax + b = c, and the other solves quadratic equations of the form ax² + bx + c = 0.

Use this page when you need a quick step-by-step solver and are deciding which algebraic form you have. It shows the rearrangement steps for linear equations and the discriminant/root calculation for quadratics, making it useful for homework checking and general algebra review.

If your problem is definitely quadratic and you also want parabola vertex, discriminant interpretation, worked examples, and a deeper formula reference, use the dedicated Quadratic Formula Calculator instead.

Solution Formulas

Linear: ax + b = c → x = (c − b) / a

Quadratic: x = (−b ± √(b²−4ac)) / (2a)

Discriminant: Δ = b² − 4ac (determines root type)

How the Equation Solver Works

Formula, assumptions, and calculation steps for this math tool.

Methodology

Math calculators apply the relevant arithmetic, algebraic, geometric, or numeric rule to the values entered and simplify the result where possible.

Calculation Steps

Read the values and operation selected.
Normalize signs, decimals, fractions, or units if needed.
Apply the mathematical rule or formula.
Format the answer and any intermediate values for checking.

Assumptions and Limits

Inputs must be within the supported domain of the operation.
Decimal answers may be rounded for readability.
Symbolic simplification is limited to the calculator scope.

Frequently Asked Questions

What is a linear equation?

A linear equation is one where the variable (x) appears only to the first power, like 2x + 5 = 13. The graph of a linear equation is always a straight line.

What is the discriminant?

The discriminant is b² − 4ac in a quadratic equation. It tells you how many real roots exist: Δ > 0 means two distinct real roots, Δ = 0 means one repeated real root, and Δ < 0 means two complex (imaginary) roots.

What are real vs complex roots?

Real roots are ordinary numbers. Complex (imaginary) roots occur when Δ < 0 and involve √(−1) = i. They always appear in conjugate pairs (a + bi and a − bi) for equations with real coefficients.

How do I check my answer?

Substitute the solution back into the original equation. For ax + b = c: compute a(x) + b and verify it equals c. For quadratics: compute ax² + bx + c and verify it equals 0.

Real-World Applications

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Break-Even Analysis

Solve a linear equation to find the production volume at which total revenue equals total cost — the break-even point.

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Projectile Motion

Quadratic equations model the height of a projectile over time — solving finds the time it hits the ground.

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Electrical Circuit Analysis

Ohm's Law and Kirchhoff's laws generate linear equations that are solved to find voltage, current, or resistance.

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Loan & Interest Calculations

Simple interest and amortisation problems reduce to linear equations relating principal, rate, and time.

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Supply & Demand Equilibrium

Setting supply and demand equations equal and solving finds the market equilibrium price and quantity.

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Chemical Equation Balancing

Stoichiometry problems form systems of linear equations — solving them ensures conservation of matter.

Common Mistakes

Sign errors when moving terms

When moving a term from one side of an equation to the other, its sign must flip — forgetting this is the most common algebraic error.

Dividing both sides by zero

If the coefficient of x is zero in a linear equation, the equation either has no solution or infinitely many — division by zero is undefined.

Using the wrong quadratic formula sign

The ± in the quadratic formula produces two roots — missing the second root (using only + or only −) gives an incomplete answer.

Ignoring complex roots

When the discriminant is negative, many students conclude "no answer" rather than correctly identifying two complex conjugate roots.

Not verifying solutions by substitution

Always substitute solutions back into the original equation to verify — particularly important when the original equation involved fractions or radicals.

Equation Types Quick Reference

Type	Standard Form	Number of Roots	Solution Method
Linear	ax + b = 0	1 real root	Isolation: x = −b/a
Quadratic (D > 0)	ax² + bx + c = 0	2 distinct real roots	Quadratic formula
Quadratic (D = 0)	ax² + bx + c = 0	1 repeated real root	Quadratic formula
Quadratic (D < 0)	ax² + bx + c = 0	2 complex roots	Quadratic formula with i
Cubic	ax³ + bx² + cx + d = 0	Up to 3 real roots	Cardano's formula / numerical

References

Stewart, James. Algebra and Trigonometry. Cengage Learning, 2015.
Axler, Sheldon. Algebra and Trigonometry. Wiley, 2021.
Larson, Ron. Precalculus: A Concise Course. Cengage, 2019.
Blitzer, Robert. Introductory and Intermediate Algebra. Pearson, 2020.
National Council of Teachers of Mathematics. Principles and Standards for School Mathematics. NCTM, 2000.