Solve quadratic equations with full steps.
x = (−b ± √(b² − 4ac)) ÷ 2a. For x² − 3x − 4 = 0 the roots are 4 and -1.
A quadratic equation has the form ax² + bx + c = 0. Enter the coefficients a, b and c and the calculator solves it with the quadratic formula, shows the discriminant and the type of roots (two real, one repeated or two complex), the vertex and axis of symmetry of the parabola, the sum and product of the roots and a step-by-step working table.
The discriminant b² − 4ac tells you the kind of roots before solving. See also the Root Calculator, Exponent Calculator, Slope Calculator and the Scientific Calculator.
The calculator applies the standard method step by step.
| Item | Value |
|---|---|
| Equation | 1x² − 3x − 4 = 0 |
| Discriminant (b² − 4ac) | 25 |
| Type of roots | Two distinct real roots |
| x₁ | 4 |
| x₂ | -1 |
| Vertex of the parabola | (1.5, -6.25) |
| Axis of symmetry | x = 1.5 |
| Opens | Upward (minimum at the vertex) |
| y-intercept | -4 |
| Sum of roots (−b/a) | 3 |
| Product of roots (c/a) | -4 |
| Step | Working |
|---|---|
| 1. Identify | a = 1, b = -3, c = -4 |
| 2. Discriminant | b² − 4ac = 9 − -16 = 25 |
| 3. Square root | √25 = 5 |
| 4. Roots | 4 and -1 |
| Equation (= 0) | Roots |
|---|---|
| x² − 3x − 4 | 4 and -1 |
| x² + 2x + 1 | -1 |
| x² + 2x + 5 | -1 ± 2i |
| 2x² − 4x − 6 | 3 and -1 |
| −x² + 4 | -2 and 2 |
x = (−b ± √(b² − 4ac)) ÷ 2a — it gives the solutions of ax² + bx + c = 0.
b² − 4ac. Positive means two real roots, zero means one repeated root, negative means two complex roots.
The parabola never crosses the x-axis; the solutions involve i, the square root of −1.
x = −b ÷ 2a; substitute that x into the equation to find y.
Then the x² term vanishes and the equation is linear, not quadratic.
Sum = −b ÷ a and product = c ÷ a — useful to check your answers.
It uses the standard method: x = (−b ± √(b² − 4ac)) ÷ 2a vertex = (−b ÷ 2a, f(−b ÷ 2a)) The full working and a worked example are shown on this page.
Decimal results use standard double-precision arithmetic (about 15–16 significant digits). Tools that work with whole numbers — big numbers, factors, bases, combinations — use exact integer arithmetic, so no digits are lost.
Yes for checking your work and learning the method. The steps and formula are shown so you can understand the answer, but follow your teacher's rules about calculators in exams.
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