Equation Solver

What an Equation Solver Does

An equation states that two expressions are equal, and solving it means finding every value of the unknown that makes the statement true. This algebra solver handles polynomial equations in one variable, x, up to the fourth power. You can type terms on both sides, in any order, with brackets and implied multiplication, and the solver does the bookkeeping that is easy to get wrong by hand: expanding brackets, collecting like terms and moving everything to one side.

Each answer comes with a check table that substitutes the root back into the original left and right sides. If the two columns agree, the root is correct. This is the same habit teachers recommend when you solve by hand, and it catches sign slips straight away.

How to Use the Calculator

Write the equation with x as the unknown and ^ for powers, so x squared is x^2. Multiplication signs are optional between a number and x or a bracket: 2x, 3(x − 1) and (x + 1)(x − 2) are all read correctly. Division is allowed by numbers, as in x/4 + 1 = 3. If you leave out the = sign, the expression is set equal to zero.

To solve simultaneous equations, use the System of Linear Equations Solver further down. Pick two or three unknowns, then enter each equation's coefficients in its own row. A system with no single answer is reported as either inconsistent (no solution) or dependent (infinitely many solutions).

How the Equation Solver Works

First both sides are expanded into polynomials and the right side is subtracted from the left, giving the standard form P(x) = 0. The degree of P decides the method. A linear equation needs one division. A quadratic uses the quadratic formula, and when the coefficients are whole numbers or short decimals the solver also gives the exact answer as a fraction or a simplified surd such as (1 ± √5)/2. If the discriminant is negative, the two roots are complex and are shown in the form p ± qi.

Worked example with the default input 3x² − 2x = x + 6. Subtracting x + 6 from both sides gives 3x² − 3x − 6 = 0, so a = 3, b = −3 and c = −6. The discriminant is (−3)² − 4 × 3 × (−6) = 9 + 72 = 81, whose square root is 9. The quadratic formula gives x = (3 ± 9) ÷ 6, so x = 2 or x = −1. Checking x = 2: the left side is 12 − 4 = 8 and the right side is 2 + 6 = 8.

ax + b = 0 → x = −b ÷ a; ax² + bx + c = 0 → x = (−b ± √(b² − 4ac)) ÷ 2a

a, b, c

the coefficients after every term is moved to one side

b² − 4ac

the discriminant, which decides how many real roots a quadratic has

Cubic and quartic equations are solved numerically: the solver finds the turning points from the derivative and then narrows down each sign change by bisection.

Cubic and Quartic Equations

Closed-form formulas exist for degree 3 and 4, but they are long and often produce real answers through complex intermediate steps. The solver instead works numerically. It finds where the derivative is zero, which splits the number line into stretches where the polynomial only rises or only falls. Each stretch can contain at most one root, and a change of sign pins it down by repeated halving to about 15 significant digits. A turning point that touches zero is reported as a repeated root with its multiplicity.

A cubic always has at least one real root because its two ends head in opposite directions. A quartic may have four, two or none. The results show how many roots are real and how many are non-real complex numbers, which always come in conjugate pairs when the coefficients are real.

Tips for Solving Equations by Hand

Using a math solver with steps is a good way to check homework, but the method is worth knowing:

• Clear fractions first by multiplying both sides by the common denominator.
• Move every term to one side before factoring; a product can only be zero if one of its factors is zero.
• Try small whole numbers such as ±1, ±2 and ±3 as roots of a cubic. Each root r you find lets you divide out a factor (x − r).
• Never divide both sides by x. It hides the root x = 0.
• Substitute your answers back into the original equation, not the rearranged one.

Frequently asked questions