Quadratic Equation Solver
By Santaji GadepatilinUpdated October 27, 2026
A quadratic equation has the form ax² + bx + c = 0. The quadratic formula gives its roots, and the discriminant tells you whether they are real or complex.
How to use the quadratic equation solver
Enter the coefficients a, b and c of ax² + bx + c = 0. The discriminant and both roots appear instantly. Click Show steps to see the formula with your numbers.
If the discriminant is negative, the roots are complex and the calculator shows a warning — there is no real x that solves the equation.
The quadratic formula
x = (−b ± √(b² − 4ac)) ÷ 2a
Worked example
2x² − 7x + 3 = 0, so a = 2, b = −7, c = 3.
- Discriminant = (−7)² − 4 × 2 × 3 = 49 − 24 = 25
- x₁ = (7 + 5) ÷ 4 = 3
- x₂ = (7 − 5) ÷ 4 = 0.5
Check: 2(3)² − 7(3) + 3 = 18 − 21 + 3 = 0 ✓
Discriminant cases
| Discriminant | Roots |
|---|---|
| Positive | Two different real roots |
| Zero | One repeated real root |
| Negative | Two complex roots (no real solution) |
Other ways to solve a quadratic
| Method | Best when |
|---|---|
| Factoring | Roots are whole numbers or simple fractions |
| Completing the square | Deriving the vertex form or the formula itself |
| Quadratic formula | Always works |
| Graphing | Seeing roughly where the roots are |
Example by factoring: 2x² − 7x + 3 = (2x − 1)(x − 3), so x = 0.5 or x = 3 — the same as the calculator.
Vertex of the parabola
The turning point of y = ax² + bx + c is at:
x = −b ÷ (2a), y = c − b² ÷ (4a)
For 2x² − 7x + 3, the vertex is at x = 1.75, y = −3.125. The parabola opens upward when a > 0 and downward when a < 0.
Real-world quadratics
- Projectile motion: height h = −4.9t² + vt + h₀ (metres, seconds). Set h = 0 to find when it lands.
- Area problems: a rectangle with perimeter 20 and area 21 gives x² − 10x + 21 = 0, so the sides are 3 and 7.
- Profit: revenue minus cost often forms a quadratic; the vertex gives the maximum profit.
Complex roots
When the discriminant is negative, the roots are complex: x = −b ÷ 2a ± i√(−D) ÷ 2a. For x² + 2x + 5 = 0, the roots are −1 ± 2i.
Sum and product of roots
For any quadratic, x₁ + x₂ = −b ÷ a and x₁ × x₂ = c ÷ a. In the example: 3 + 0.5 = 3.5 = 7/2 and 3 × 0.5 = 1.5 = 3/2.
Frequently asked questions
What is the quadratic formula?
x = (−b ± √(b² − 4ac)) ÷ 2a. It gives both solutions of ax² + bx + c = 0.
What does the discriminant tell me?
If it is positive there are two real roots; if zero, one repeated root; if negative, no real roots (the roots are complex).
What if a = 0?
Then it is not quadratic — it is the linear equation bx + c = 0, with the single solution x = −c ÷ b.
How do I find the roots of x² − 5x + 6 = 0?
a = 1, b = −5, c = 6. The discriminant is 1, so x = (5 ± 1) ÷ 2, giving 3 and 2.
What is the vertex form of a quadratic?
y = a(x − h)² + k, where (h, k) is the vertex. It comes from completing the square.