Overview
A quadratic equation has the form ax² + bx + c = 0 where a is not zero. It can be solved by factoring, completing the square, or the quadratic formula, which handles every case including complex roots.
Formula
x = (−b ± √(b² − 4ac)) ÷ 2a, with discriminant Δ = b² − 4ac. The sum of roots is −b/a and their product is c/a.
How it works
Identify the coefficients
Write the equation in standard form and note a, b and c with their signs.
Compute the discriminant
Δ = b² − 4ac tells you the root type before solving.
Apply the quadratic formula
Substitute into x = (−b ± √Δ) ÷ 2a to get both roots.
Verify and read the vertex
Substitute the roots back, and note the parabola vertex at x = −b/2a.
Worked examples
Example 1: x² − 3x + 2 = 0
Δ = 9 − 8 = 1, so x = (3 ± 1) ÷ 2, giving roots 2 and 1.
Example 2: x² + 2x + 5 = 0
Δ = 4 − 20 = −16, so the roots are complex: −1 ± 2i.
Frequently asked questions
What does a negative discriminant mean?
There are no real roots, but there are two complex conjugate roots.
Can every quadratic be solved by factoring?
No. Factoring works when the roots are simple, while the quadratic formula works in all cases.
Why is the vertex useful?
It gives the highest or lowest point of the parabola, which is useful for maximum and minimum problems.
This calculator is for educational and informational purposes only and is not a substitute for professional advice.