Solve Any Quadratic Equation
A quadratic equation has the form ax² + bx + c = 0, where a is not zero. Its graph is a U-shaped curve called a parabola. This quadratic equation solver finds both roots (real or complex), the discriminant, the vertex of the parabola and its axis of symmetry. Formulas are rendered clearly so you can follow the math.
The Quadratic Formula
x = (−b ± √(b² − 4ac)) ÷ 2a. The expression under the square root, b² − 4ac, is called the discriminant.
Example: x² − 5x + 6 = 0 has a = 1, b = −5 and c = 6. The discriminant is 25 − 24 = 1, so x = (5 ± 1) ÷ 2, which gives x = 3 or x = 2. The vertex is at x = −b ÷ 2a = 2.5, y = −0.25.
What the Discriminant Tells You
- Discriminant > 0: two different real roots (the parabola crosses the x-axis twice).
- Discriminant = 0: one repeated real root (the parabola touches the x-axis at its vertex).
- Discriminant < 0: two complex roots (the parabola never crosses the x-axis).
Frequently Asked Questions
What are complex roots?
When the discriminant is negative, the square root of a negative number produces imaginary numbers written with i, where i = √−1. The roots come as a pair like 2 + 3i and 2 − 3i.
What is the vertex of a parabola?
The vertex is the highest or lowest point of the parabola. Its x-coordinate is −b ÷ 2a. If a is positive, the vertex is a minimum; if a is negative, it is a maximum.
Can I use this if my equation isn't equal to zero?
Yes, but first rearrange it so one side equals zero. For example, x² + 3x = 10 becomes x² + 3x − 10 = 0, so a = 1, b = 3 and c = −10.