Quadratic Formula Calculator

How this is calculated

The discriminant (b²-4ac) is calculated first to determine the type of roots, then the quadratic formula is applied to find the exact value(s) of x.

The quadratic formula

Any equation in standard form ax² + bx + c = 0 (with a ≠ 0) can be solved using the quadratic formula, regardless of whether it factors nicely.

Formula

Example: for x²-3x-4=0 (a=1, b=-3, c=-4), the discriminant is 9+16=25, giving x = (3±5)/2, so x = 4 or x = -1.

Quick reference table

Common pitfalls

• Sign errors with b — when b is negative, -b becomes positive; it's a common place to drop a sign.
• Forgetting the ± symbol — a quadratic generally has two solutions, not one, unless the discriminant is exactly zero.
• Misidentifying a, b, c — the equation must be in standard form (set equal to zero) before reading off the coefficients.

Tips

• If a term is missing (like no bx term), its coefficient is simply zero — plug in 0 for that value.
• Checking your answer by substituting it back into the original equation is a reliable way to catch arithmetic mistakes.

Frequently asked questions

What is the quadratic formula? x = (-b ± √(b²-4ac)) ÷ 2a, used to solve any equation in the form ax² + bx + c = 0.

What is the discriminant and what does it tell you? The discriminant is b²-4ac. If it's positive, the equation has two real roots; if zero, one repeated real root; if negative, two complex roots.

What does it mean for a quadratic to have complex roots? Complex roots occur when the discriminant is negative, meaning the parabola never crosses the x-axis — the solutions involve the imaginary unit i (√-1).

Can the quadratic formula be used for any quadratic equation? Yes, as long as the equation is in standard form ax² + bx + c = 0 with a not equal to zero — the quadratic formula always works, unlike factoring, which only works cleanly for some equations.

What is an example of solving with the quadratic formula? For x²-3x-4=0 (a=1, b=-3, c=-4), the discriminant is 9+16=25, giving x = (3±5)/2, so x = 4 or x = -1.

Worked examples

Example 1 — Two real roots: x²-3x-4=0 → x = 4 or x = -1.

Example 2 — One repeated root: x²-4x+4=0 → discriminant = 0 → x = 2.

Example 3 — Complex roots: x²+2x+5=0 → discriminant = -16 → x = -1 ± 2i.