How to use the Algebra Calculator
- Enter the equation with the unknown written as x.
- Use * for multiplication and ^ for powers, for example 3*x^2 - 5*x = 2.
- Read the solution set, including both roots for quadratics.
- Substitute your answer back into the original equation to confirm it.
How the calculation works
Linear equations reduce by collecting like terms on one side and constants on the other, then dividing by the coefficient of x — every operation applied identically to both sides, which is the single rule that keeps the statement true. A linear equation has exactly one solution unless the x terms cancel, which signals either no solution or infinitely many.
Quadratics are rearranged to ax² + bx + c = 0 and solved with the quadratic formula. The discriminant b² − 4ac decides the shape of the answer: positive gives two distinct real roots, zero gives one repeated root where the parabola touches the axis, negative gives a conjugate pair of complex roots. Checking by substitution catches sign slips, which are by far the most common error.
Linear: x = (c − b) / a ; Quadratic: x = (−b ± √(b² − 4ac)) / 2aSource: Standard algebraic balancing method and the quadratic formula derived by completing the square.
Worked example
Solve 2x² − 7x − 15 = 0.
- a = 2, b = −7, c = −15.
- Discriminant = 49 − 4(2)(−15) = 49 + 120 = 169.
- √169 = 13, so x = (7 ± 13) / 4.
- x = 5 or x = −1.5.
Two real roots: x = 5 and x = −1.5. Substituting 5 gives 50 − 35 − 15 = 0, confirming the answer.
Frequently asked questions
What does a negative discriminant mean?+
There are no real solutions — the parabola never crosses the x-axis. The two roots are complex conjugates.
Can I always factorise instead?+
Only when the roots are rational. The quadratic formula works in every case, so it is the safer default.
Why check the answer?+
Substitution catches arithmetic and sign errors instantly, and it is the only verification that does not repeat your original reasoning.
What if the x terms cancel out?+
If you are left with a true statement there are infinitely many solutions; a false one means no solution exists.
Last reviewed August 31, 2026. We review this page whenever the underlying formula, tax year, published rate or standard changes.