Quadratic Formula Calculator
Solve any quadratic equation ax² + bx + c = 0. Enter the three coefficients to get both solutions — real or complex — along with the discriminant, the type of roots and the vertex of the parabola.
- x₁
- 3
- x₂
- 2
- Discriminant (b² − 4ac)
- 1
- Type of roots
- Two distinct real roots
- Vertex
- (2.5, -0.25)
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Formula
How to use it
- Rearrange your equation so one side is zero: ax² + bx + c = 0.
- Enter a, b and c, keeping their signs. Use 0 for a missing term.
- Read the solutions. The discriminant line tells you why there are two, one or no real roots.
Worked examples
x² − 5x + 6 = 0
- Solutions
- x = 3 or x = 2
- Discriminant (b² − 4ac)
- 1
- Type of roots
- Two distinct real roots
- Vertex
- (2.5, -0.25)
x² − 4x + 13 = 0, which has no real solutions
- Solutions
- x = 2 + 3i or x = 2 - 3i
- Discriminant (b² − 4ac)
- -36
- Vertex
- (2, 9)
x² + 6x + 9 = 0
- Solutions
- x = -3 (double root)
- Discriminant (b² − 4ac)
- 0
What the discriminant tells you
If D is positive there are two different real solutions and the parabola crosses the x-axis twice. If D is zero there is one repeated solution and the parabola just touches the axis at its vertex. If D is negative there are no real solutions; the two roots are complex conjugates p ± qi, where p = −b ÷ 2a and q = √(−D) ÷ 2a.
A worked example and a quick check
For x² − 5x + 6 = 0: D = 25 − 24 = 1, so x = (5 ± 1) ÷ 2, giving x = 3 and x = 2.
To check any answer, the two roots must add up to −b ÷ a and multiply to c ÷ a. Here 3 + 2 = 5 and 3 × 2 = 6.
Questions people ask
What are the solutions of x² − 5x + 6 = 0?
x = 3 and x = 2. The equation factors as (x − 3)(x − 2) = 0.
What happens when the discriminant is negative?
There are no real solutions, only a pair of complex ones. For x² − 4x + 13 = 0 the discriminant is −36 and the solutions are x = 2 + 3i and x = 2 − 3i.
Why can a not be zero?
With a = 0 the x² term disappears and the equation is linear, bx + c = 0, with the single solution x = −c ÷ b. The quadratic formula would divide by zero.