Quadratic Equation Calculator
Solve ax² + bx + c = 0 with every step of the quadratic formula, including complex roots and the turning point.
x₁
5
- x₂
- -2
- Discriminant (b² − 4ac)
- 49
- Vertex (turning point)
- (1.5, -12.25)
1x² − 3x − 10 = 0 has two real roots, x = 5 and x = -2. The curve turns at (1.5, -12.25).
How it's worked out
- Equation1x² − 3x − 10 = 0
- Discriminant
(-3)² − 4 × 1 × -1049 - Quadratic formula
x = (−b ± √49) ÷ 25 and -2 - Vertex
x = −b ÷ 2a = 1.5(1.5, -12.25)
How it is solved
Discriminant D = b² − 4ac · x = (−b ± √D) ÷ 2a
Worked example
For x² − 3x − 10 = 0, D = 9 + 40 = 49, so x = (3 ± 7) ÷ 2, giving x = 5 and x = −2. The vertex is at (1.5, −12.25).
Questions people ask
What is the quadratic formula?
x = (−b ± √(b² − 4ac)) ÷ 2a. The part under the square root is the discriminant.
What does the discriminant tell me?
Above zero, two real roots; exactly zero, one repeated root; below zero, no real roots but two complex ones.
What is the vertex?
The turning point of the curve, at x = −b ÷ 2a. It's the lowest point when a is positive and the highest when a is negative.
About this calculator
Formula and rules last checked 25 October 2026Written and tested by the CalcPenguin teamReport a problem