Quadratic Formula Calculator
Solve ax² + bx + c = 0 in one step. The tool applies the quadratic formula, reports the discriminant so you can see why there are zero, one, or two real solutions, and handles complex roots when the parabola never touches the x-axis. Below the calculator you'll find the step-by-step method, a comparison of solving techniques, and real-world examples.
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How to Solve a Quadratic Equation Step by Step
Follow these five steps for any equation you can write as ax² + bx + c = 0:
- Identify a, b, c. Match your equation to the standard form. For
2x² − 4x − 6 = 0, a = 2, b = −4, c = −6. - Compute the discriminant D = b² − 4ac. This single number predicts how many real roots you'll get.
- If D ≥ 0, plug into
x = (−b ± √D) ÷ (2a). The ± gives the two candidates (or one if D = 0). - If D < 0, the roots are complex:
x = −b/(2a) ± (√−D)/(2a)·i. There is no real solution, only a conjugate pair. - Check your work by substituting each root back into the original equation — it should simplify to 0.
The Quadratic Formula and the Discriminant
For ax² + bx + c = 0:
x = (−b ± √(b² − 4ac)) ÷ (2a)
The discriminant (b² − 4ac) is the decision-maker. Positive → two real roots (the parabola crosses the axis twice); zero → one root (it just touches); negative → no real roots, only a complex pair. This is the same algebra that underlies projectile motion, optimization problems, and many growth models.
Quadratic Formula vs. Factoring vs. Completing the Square
Four ways to solve a quadratic — each has strengths. The formula always works; the others are faster only in special cases.
Which solving method should you use?
| Quadratic FormulaBest always works | Factoring special cases | Complete Square derivation | Graphing visual |
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When in doubt, use the quadratic formula — it is the only method guaranteed to find every root, real or complex.
Worked Example: x² − 5x + 6 = 0
With a=1, b=−5, c=6: the discriminant is (−5)² − 4·1·6 = 25 − 24 = 1 (positive, so two roots). x = (5 ± √1) ÷ 2 = (5 ± 1) ÷ 2, giving x = 3 and x = 2. You can check by factoring: (x − 2)(x − 3) = x² − 5x + 6. For a negative discriminant, say x² + 1 = 0, the roots are 0 ± 1i — no real solution.
Common Mistakes to Avoid
- Forgetting the ±. The formula gives two roots (unless D = 0); write both.
- Sign of −b. If b is negative, −b becomes positive — easy to mistype.
- a = 0. Then it's linear, not quadratic; the formula divides by zero.
- Negative discriminant panic. D < 0 isn't "no answer" — it's two complex answers. Report them as a ± bi.
Frequently Asked Questions
What is the quadratic formula?
x = (−b ± √(b²−4ac)) ÷ (2a). The discriminant b²−4ac sets the root count.
What does the discriminant tell you?
Positive → 2 real roots; zero → 1 root; negative → complex pair, no real roots.
Can a quadratic have no real solution?
Yes — a negative discriminant means complex roots (a ± bi), e.g. x² + 1 = 0 → ±i.
Why must a not be zero?
If a = 0 it's linear (bx + c = 0), and the formula divides by 2a, so it's undefined.
How do I solve step by step?
Identify a,b,c → compute D = b²−4ac → if D≥0 use x=(−b±√D)/(2a), else use the complex form → check by substitution.