Quadratic Formula Calculator

Solve any equation in the form ax² + bx + c = 0 for real or complex roots, with the discriminant and every step shown.

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Enter values and press Calculate to see the result here.

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

x = (-b ± √(b²-4ac)) ÷ 2a

Discriminant = b² - 4ac

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

Discriminant	Root type
Positive	Two distinct real roots
Zero	One repeated real root
Negative	Two complex conjugate roots

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.

Where quadratics show up and which method to use

Choosing a solving method

Method	Best when
Factoring	The numbers are small and the equation factors neatly.
Square roots	There is no middle term, such as x² = 49.
Completing the square	You also need the vertex form of the parabola.
Quadratic formula	It always works, including when roots are irrational or complex.

Useful facts about the roots

The sum of the roots equals −b ÷ a.
The product of the roots equals c ÷ a.
The vertex of the parabola has an x-coordinate of −b ÷ 2a.
If a is positive the parabola opens upward and has a minimum; if negative it opens downward and has a maximum.

Real-world uses

Quadratic equations describe the height of a thrown object over time, the area of a rectangle when one side depends on the other, and the profit of a business when price and demand interact. In each case the roots tell you when a value reaches zero, such as when a ball lands, and the vertex tells you the highest or lowest point.