📐 Quadratic Equation Solver
Solve any quadratic equation ax² + bx + c = 0 with the quadratic formula – real or complex roots, discriminant and vertex.
Quick answer: For x² − 3x + 2 = 0 the discriminant is (−3)² − 4 × 1 × 2 = 1, so the quadratic formula gives x = (3 ± 1) ÷ 2: the roots are x = 2 and x = 1, and the vertex is (1.5, −0.25).
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Quadratic Equation Solver inputs
Result
x² − 3x + 2 = 0
x₁ = 2, x₂ = 1
Two distinct real roots (discriminant > 0)
- Discriminant (b² − 4ac)
- 1
- Vertex
- (1.5, −0.25)
- Axis of symmetry
- x = 1.5
| Parabola opens | Upward (minimum at vertex) |
| y-intercept | (0, 2) |
| Sum of roots (−b/a) | 3 |
| Product of roots (c/a) | 2 |
Step by step
- Identify the coefficients: a = 1, b = −3, c = 2
- Discriminant: D = b² − 4ac = (−3)² − 4 × 1 × 2 = 1
- Quadratic formula: x = (−b ± √D) ÷ 2a = (3 ± √1) ÷ 2
- Roots: x₁ = 2, x₂ = 1
- Vertex: x = −b ÷ 2a = 1.5, y = c − b² ÷ 4a = −0.25
The quadratic formula
Any equation of the form ax² + bx + c = 0 with a ≠ 0 can be solved with one formula:
x = ( −b ± √(b² − 4ac) ) ÷ 2a
The expression under the square root, D = b² − 4ac, is the discriminant. It tells you what kind of solutions to expect before you finish the calculation.
| Discriminant | Roots | Graph |
|---|---|---|
| D > 0 | Two distinct real roots | Crosses the x-axis twice |
| D = 0 | One repeated real root | Touches the x-axis at the vertex |
| D < 0 | Two complex conjugate roots | Never meets the x-axis |
Worked example
Solve x² − 3x + 2 = 0. Here a = 1, b = −3 and c = 2.
- D = (−3)² − 4 × 1 × 2 = 9 − 8 = 1.
- x = (3 ± √1) ÷ 2 = (3 ± 1) ÷ 2.
- x₁ = 4 ÷ 2 = 2 and x₂ = 2 ÷ 2 = 1.
Check by factoring: x² − 3x + 2 = (x − 1)(x − 2). The vertex sits halfway between the roots at x = 1.5, where y = −0.25.
Vertex and symmetry
The parabola y = ax² + bx + c turns at x = −b ÷ 2a. Substituting back gives the y-coordinate, c − b² ÷ 4a. The sum of the roots always equals −b/a and their product equals c/a – a quick way to check your answers.
Tips
- Rearrange the equation so one side is zero before reading off a, b and c.
- Watch the signs: if the equation has “− 5x”, then b = −5.
- Try factoring first when the numbers are small; use the formula when they are not.
- The solver uses a numerically stable form of the formula, so it stays accurate even when b² is much larger than 4ac.
Frequently asked questions
What does the discriminant tell you?
The discriminant D = b² − 4ac decides the type of roots. If D > 0 there are two different real roots, if D = 0 there is one repeated real root, and if D < 0 there are two complex conjugate roots and the parabola never crosses the x-axis.
How do I enter an equation like 2x² = 8?
Move every term to one side first so it reads 2x² − 8 = 0. Then a = 2, b = 0 and c = −8, and the roots are x = 2 and x = −2.
What are complex roots?
When the discriminant is negative, the square root of D is imaginary. The roots then have the form p ± qi, where i = √−1. For x² + 2x + 5 = 0 they are −1 + 2i and −1 − 2i.
What is the vertex of a parabola?
It is the turning point of y = ax² + bx + c, located at x = −b / 2a. It is the minimum when a > 0 and the maximum when a < 0, and the graph is symmetric about the vertical line through it.