Quadratic Formula Calculator: Solve ax² + bx + c = 0
The quadratic formula is one of the most memorized things in high school math — and one of the first things people forget once they're done with school. Here's how it works and when you need it.
The Formula
For any equation in the form ax² + bx + c = 0, the solutions are:
x = (−b ± √(b²−4ac)) / 2a
The ± means there are usually two answers — you solve it once with + and once with −. The calculator above shows both roots, the discriminant value, and the steps.
What the Discriminant Tells You
Before solving, the discriminant (b²−4ac) tells you what kind of answer to expect:
| Discriminant | Number of solutions | Type |
|---|---|---|
| Positive (> 0) | Two | Distinct real roots |
| Zero (= 0) | One | Repeated real root |
| Negative (< 0) | None real | Complex conjugate pair |
Reading the Results
Two real roots — the most common case. The parabola crosses the x-axis at two points. Example: x² − 5x + 6 = 0 → x = 3 and x = 2.
One repeated root — the parabola just touches the x-axis at one point, like a ball bouncing at the bottom. Example: x² − 6x + 9 = 0 → x = 3 (twice).
Complex roots — the parabola never reaches the x-axis. This isn't an error; it means there are no real solutions. The complex roots come in pairs like 2 + 3i and 2 − 3i. These are useful in engineering, signal processing, and physics — but not in most everyday math problems.
When a = 0
If the x² term is zero, you no longer have a quadratic — it's a linear equation (bx + c = 0). The calculator handles this case and solves it as x = −c/b. If both a and b are zero, the equation either has every real number as a solution (0 = 0) or no solution at all.
Frequently Asked Questions
What does the discriminant tell you?
The discriminant is b²−4ac, the expression under the square root in the quadratic formula. If it's positive, there are two distinct real roots. If it's zero, there's exactly one real root (a repeated root). If it's negative, there are no real solutions — only complex (imaginary) roots.
What are complex roots and do they matter?
Complex roots appear when the discriminant is negative, meaning the parabola never crosses the x-axis. They come in conjugate pairs like 3 + 2i and 3 − 2i. In pure math and engineering (signal processing, electrical circuits) complex roots are very meaningful. For most everyday problem-solving, a negative discriminant just means the equation has no real solutions.
What if a equals zero?
If a = 0, the equation is no longer quadratic — it reduces to a linear equation bx + c = 0. The calculator handles this: if a = 0 and b ≠ 0, it solves the linear case. If both a = 0 and b = 0, the equation is either always true (if c = 0) or has no solution (if c ≠ 0).
How do I check my answer?
Plug your root back into the original equation. If x₁ is a root of ax² + bx + c = 0, then a(x₁)² + b(x₁) + c should equal exactly zero. For x² − 5x + 6 = 0 with roots 3 and 2: (3)² − 5(3) + 6 = 9 − 15 + 6 = 0. ✓
This article is for informational purposes only. See our disclaimer.