Quadratic Equation Solver
A quadratic equation has the form ax² + bx + c = 0 with a ≠ 0. It can have two real solutions, one repeated solution or two complex solutions. This solver applies the quadratic formula, tells you which case you are in through the discriminant, and also reports the vertex of the parabola y = ax² + bx + c.
How to use the quadratic equation solver
- Rearrange your equation into the form ax² + bx + c = 0.
- Enter a, b and c. Use a negative sign for negative coefficients and 0 when a term is missing.
- Read the roots, the discriminant and the vertex. Complex roots are shown in the form p ± qi.
Formula
x = (−b ± √(b² − 4ac)) ÷ 2a
The discriminant D = b² − 4ac decides the case: D > 0 gives two real roots, D = 0 gives one repeated root, and D < 0 gives two complex roots. The vertex is at x = −b ÷ 2a.
Worked examples
Two real roots
x² − 5x + 6 = 0 has a = 1, b = −5, c = 6. D = 25 − 24 = 1, so x = (5 ± 1) ÷ 2, giving x = 3 and x = 2. Check: 3² − 15 + 6 = 0.
Complex roots
x² + 2x + 5 = 0 has D = 4 − 20 = −16. The roots are −1 + 2i and −1 − 2i. The parabola never crosses the x-axis.
A larger coefficient
2x² − 4x − 6 = 0 has D = 16 + 48 = 64, so x = (4 ± 8) ÷ 4, giving x = 3 and x = −1, with the vertex at (1, −8).
Three ways to solve a quadratic
The quadratic formula always works, but it is not the only method. Knowing all three helps you choose the fastest one and check your answers.
1. Factoring
If the equation factors neatly, you can read the roots straight off. x² − 5x + 6 = (x − 2)(x − 3), so x = 2 or x = 3. Look for two numbers that multiply to c and add to b. This is quick when the numbers are small integers.
2. Completing the square
Rewrite ax² + bx + c as a(x + h)² + k. Solving x² + 6x + 5 = 0: x² + 6x + 9 = 4, so (x + 3)² = 4 and x + 3 = ±2, giving x = −1 or x = −5. This method is how the quadratic formula is derived, and it also gives the vertex form of a parabola.
3. The quadratic formula
Use it whenever factoring is not obvious, or when the coefficients are decimals or fractions. It is a formula, so you cannot fail to find an answer; you only have to be careful with signs.
Reading the graph
The roots are where the parabola crosses the x-axis. Two real roots mean two crossings, a repeated root means the parabola just touches the axis at its vertex, and complex roots mean it floats above or below the axis. The sign of a tells you whether the parabola opens upward (a > 0) or downward (a < 0).
Where quadratics appear
Projectile motion (the height of a ball over time), areas (finding a side length from an area), profit models with a maximum, and the curved cables of suspension bridges all use quadratics.
Practice problems
Try these yourself first, then check your answer. The answers come from the calculator above.
- Solve x² − 7x + 10 = 0.
Show answer
x₁: 5 · x₂: 2
- Solve 3x² + 2x − 8 = 0.
Show answer
x₁: 1.333333333 · x₂: -2
- Solve x² + 4x + 13 = 0.
Show answer
x₁: -2 + 3i · x₂: -2 − 3i
Common mistakes to avoid
- Forgetting that b² is positive even when b is negative: (−5)² = 25.
- Dividing by 2 instead of 2a in the denominator.
- Not moving every term to one side first; the right-hand side must be 0.
Frequently asked questions
What is the discriminant?
It is b² − 4ac, the part under the square root in the quadratic formula. Its sign tells you how many real solutions the equation has.
Can a quadratic equation have no solution?
It always has solutions among the complex numbers. It has no real solution when the discriminant is negative.
What if a is 0?
Then the equation is linear (bx + c = 0) with the single solution x = −c ÷ b, not a quadratic.
What is the vertex used for?
It is the turning point of the parabola: the minimum when a > 0 and the maximum when a < 0.
Reviewed September 26, 2026 by the SolveCalcPro editorial team. Found a mistake? Tell us; see our editorial policy.