Equation solver
Choose the type of equation and type the coefficients straight into it. The step-by-step solution appears underneath, the way you’d write it in class.
Type the coefficients. An empty box means 0. Fractions and roots work too, such as 1/2 or √2.
Check the marked boxes: enter a number such as −3, 2.5 or 1/2.
Solution
Step-by-step solution
Read off the coefficients.
a = 2, b = −3, c = −5
Calculate the discriminant.
Δ = b² − 4ac = (−3)² − 4 · 2 · (−5) = 9 − (−40) = 49
Δ > 0, so the equation has two solutions.
√Δ = √49 = 7
Calculate the roots.
x₁ = −b − √Δ2a = 3 − 74 = −1
x₂ = −b + √Δ2a = 3 + 74 = 52 = 2.5
Factored form
2(x + 1)(x − 52) = 0
Vertex of the parabola and vertex form
p = −b2a = 34, q = −Δ4a = −498
V = (34, −498), y = 2(x − 34)² − 498
What the solver handles
- Linear equations of the form ax + b = cx + d, including identities (every number is a solution) and inconsistent equations (no solution).
- Quadratic equations ax² + bx + c = 0: discriminant, exact roots, factored form, vertex and vertex form.
- Cubic equations ax³ + bx² + cx + d = 0: rational root, synthetic division and repeated roots.
- Systems of 2 or 3 linear equations with determinants (Cramer’s rule).
Quadratic formulas
| Discriminant | Δ = b² − 4ac |
|---|---|
| Two solutions (Δ > 0) | x₁ = (−b − √Δ) / 2a, x₂ = (−b + √Δ) / 2a |
| One solution (Δ = 0) | x₀ = −b / 2a |
| Vertex | V = (−b / 2a, −Δ / 4a) |
| Vieta’s formulas | x₁ + x₂ = −b / a, x₁ · x₂ = c / a |
See it on a graph opens the function plotter with your equation: the zeros of the graph are the solutions.
Common questions
How do I solve a quadratic equation?
Calculate the discriminant Δ = b² − 4ac. If Δ > 0 there are two solutions, x₁ = (−b − √Δ)/2a and x₂ = (−b + √Δ)/2a. If Δ = 0 there is one solution, x₀ = −b/2a. If Δ < 0 there are no real solutions.
What if the discriminant is negative?
Then the equation has no real solutions. The solver also shows the two complex solutions, which you need in higher maths and electrical engineering.
How do I solve a cubic equation?
First look for a rational root among the divisors of the constant term divided by the divisors of the leading coefficient. Dividing the polynomial by that factor leaves a quadratic. When there is no rational root, the solver calculates the roots numerically.
How does Cramer’s rule work?
Calculate the main determinant D and the determinants Dx and Dy, where the variable’s column is replaced by the constants. If D ≠ 0, x = Dx/D and y = Dy/D. If D = 0, the system has either infinitely many solutions or none.
Can I use fractions as coefficients?
Yes. Type 1/2, −3/4 or even √2. When every coefficient is rational, the answer is exact, as a fraction or with a square root, with the decimal value next to it.