Quadratic Formula Calculator

Enter a, b, and c for ax² + bx + c = 0. You get the discriminant, the steps, and both roots.

Smart Tips

• This page solves ax² + bx + c = 0 only. If a is 0, the equation is not degree 2.

• The discriminant is b² − 4ac. Its sign tells you what kind of roots to expect.

• A positive discriminant gives two real roots. Zero gives one repeated real root.

• A negative discriminant is not left as a square root of a negative. The roots are labeled complex.

The Formula

x = (−b ± √(b² − 4ac)) / (2a)

a = Quadratic term
Must be non-zero

b = Linear term
Coefficient of x

c = Constant
The number with no x

D = Discriminant
b² − 4ac

If D > 0, there are two real roots. If D = 0, there is one repeated real root, and both root lines show that value. If D < 0, the roots are real-part ± imaginary-part i, and the page labels them complex.

Key Insights

Degree 2 only

The formula needs a non-zero a. A linear equation is rejected instead of being forced through the quadratic formula.

Discriminant first

The page computes D before the roots. You can see why the roots are real, repeated, or complex.

Complex roots

When D is negative, √D is not a real number. The page writes p + qi and p − qi, with q positive, and says the roots are complex.

Steps stay visible

Each calculation lists a, b, and c, the discriminant arithmetic, and the substitution into the formula.

Frequently Asked Questions

Only ax² + bx + c = 0, with a not equal to zero. It does not solve cubic equations or a general expression from the scientific keypad.

D = b² − 4ac. If D is positive, there are two different real roots. If D is zero, there is one repeated real root. If D is negative, there are two complex roots.

They are written as a real part plus or minus an imaginary part times i, for example −1 + 2i and −1 − 2i. The status line says “Two complex roots.” The page does not leave a negative number under a radical.

The plus-or-minus in the formula gives the same number twice. Showing both lines makes that repeated root obvious. For x² − 2x + 1 = 0, both roots are 1.

A negative a is still degree 2. The formula divides by 2a, so the signs follow the algebra. Only a = 0 is rejected.

The quadratic formula and the discriminant, including the case of two complex solutions when the discriminant is negative, follow OpenStax College Algebra 2e, section 2.5. The same formula is derived in OpenStax Elementary Algebra 2e, section 10.3, which stops at real solutions. No outside data feed is used.

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Discriminant

Enter values to calculate

Root 1

Root 2

Note:Degree 2 only. A negative discriminant is labeled as two complex roots, not as a real square root of a negative number. A coefficient a of zero is rejected.