Quadratic Formula Calculator
Solve any quadratic equation ax² + bx + c = 0 for real or complex roots using the quadratic formula.
| Term | Value |
|---|---|
| b² | 9 |
| 4ac | 8 |
| Discriminant (b² − 4ac) | 1 |
| √Discriminant | 1 |
| x₁ = (−b + √disc) / 2a | 2 |
| x₂ = (−b − √disc) / 2a | 1 |
A positive discriminant means the parabola crosses the x-axis twice — two distinct real roots.
- x₂1
ƒShow your work
x = [−b ± √(b²−4ac)] / 2a
- 1Discriminant = -3² − 4×1×2 = 1
About the Quadratic Formula Calculator
The quadratic formula is the tool for when factoring an equation like ax² + bx + c = 0 by inspection just isn't happening — the roots aren't nice round numbers, or there's no obvious pair of factors to spot. It shows up constantly in algebra coursework, but also anywhere a relationship is genuinely quadratic: a ball's height over time under gravity, the break-even point of a cost curve with a squared term, or an engineer solving for a dimension in a design equation that reduces to this exact form.
This calculator takes the three coefficients a, b and c and returns both roots, handling all three cases the formula can produce — two distinct real roots, one repeated real root, or a pair of complex roots when the equation never actually crosses the x-axis. It's a fast way to check an answer worked out by hand, or to solve an equation that's too messy to factor confidently.
How it’s calculated
The quadratic formula, x = [−b ± √(b² − 4ac)] / 2a, comes from completing the square on the general equation ax² + bx + c = 0. It always produces exactly two solutions because of the ± — one from adding the square root, one from subtracting it.
The expression under the square root, b² − 4ac, is called the discriminant, and its sign alone tells you what kind of roots to expect before you even finish the calculation. A positive discriminant means two distinct real roots (the parabola crosses the x-axis twice). A discriminant of exactly zero means one repeated real root (the parabola just touches the x-axis at its vertex). A negative discriminant means the square root of a negative number, which produces two complex roots — the parabola never touches the x-axis at all.
Frequently asked questions
What does the discriminant tell you?
Its sign predicts the type of roots without solving the whole equation: positive means two real roots, zero means one repeated real root, and negative means two complex (non-real) roots.
Can a quadratic equation have no real solutions?
Yes — whenever the discriminant b² − 4ac is negative. The parabola described by the equation never crosses the x-axis, so the solutions exist only as complex numbers.
What happens if a is 0?
The equation stops being quadratic and becomes linear (bx + c = 0), which the quadratic formula can't handle since it would require dividing by 2a = 0. Solve bx + c = 0 directly instead: x = −c/b.
Is the quadratic formula better than factoring?
Factoring is faster when the roots are simple integers or fractions you can spot by inspection, but the quadratic formula always works, even when the roots are irrational, complex, or otherwise messy — it's the reliable fallback when factoring isn't obvious.
How are the two roots related to a, b and c?
Their sum always equals −b/a and their product always equals c/a, regardless of whether the roots are real or complex. This is a handy way to double-check a quadratic formula result by hand.
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