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.
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The Quadratic Formula
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.
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.
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.