Algebra Equation Solver
Solve linear, quadratic, and 2-equation systems with full step-by-step working. Supports ax + b = c, ax² + bx + c = 0, and {ax+by=c, dx+ey=f}.
What is Algebra Equation Solver?
The algebra equation solver handles three of the most common algebra problem types: linear equations in one variable, quadratics (using the discriminant), and 2x2 systems of linear equations (using Cramer's rule).
For every problem it shows step-by-step work plus a verification step you can plug back into the original equation to double-check.
Formula
Linear (ax + b = c): x = (c − b) / a.
Quadratic (ax² + bx + c = 0): x = (−b ± √(b² − 4ac)) / (2a). The discriminant Δ = b² − 4ac tells you what kind of roots: positive = two real roots, zero = one repeated root, negative = two complex roots.
System (ax+by=c, dx+ey=f): using determinants — det = ae − bd, x = (ce − bf)/det, y = (af − cd)/det. If det = 0, the lines are parallel or identical.
Worked example
Solve x² − 3x + 2 = 0:
- a = 1, b = −3, c = 2
- Δ = (−3)² − 4(1)(2) = 9 − 8 = 1 (positive: two real roots)
- x = (3 ± 1) / 2 → x = 2 or x = 1
Solve 2x + 3y = 13, 4x − y = 5:
- det = 2(−1) − 3(4) = −14
- x = (13(−1) − 3(5)) / (−14) = −28 / −14 = 2
- y = (2(5) − 13(4)) / (−14) = −42 / −14 = 3
How to use this calculator
- Pick the equation type from the dropdown.
- Enter the coefficients (a, b, c — and d, e, f for systems).
- The solver returns the solution, step-by-step working, the discriminant (for quadratics), and a verification you can use to check by hand.
Frequently asked questions
How do I read the discriminant?
For ax² + bx + c = 0, the discriminant Δ = b² − 4ac. If Δ > 0 there are two real roots. Δ = 0 means one repeated real root. Δ < 0 means two complex roots (no real solutions).
Why does my system have "no unique solution"?
If the determinant ae − bd equals zero, the two equations describe either parallel lines (no intersection) or the same line (infinite intersections). Either way, there's no single (x, y) that uniquely solves both.
Can it handle equations with x on both sides?
Rewrite first. For example, 2x + 3 = 5x − 7 becomes 2x − 5x = −7 − 3, then −3x = −10, then linear with a = −3, b = 0, c = −10 (or simpler: a = −3, b = 10 in ax + b = 0 form).
What about higher-degree polynomials (cubic, quartic)?
This solver covers up to degree 2. For cubics or higher, you need numerical methods (Newton's method) or more advanced symbolic tools. Try our quadratic equation solver or graph the function to find roots visually.