Quadratic Formula Calculator

What this calculator does

The Quadratic Formula Calculator solves equations of the form ax² + bx + c = 0. It reports the discriminant, the roots, and the steps used to find them. Depending on the discriminant, the equation may have two real roots, one repeated real root, or a pair of complex roots.

Quadratic equations appear in projectile motion, area problems, optimization, finance, and many branches of science. The quadratic formula is a general method that works for every quadratic equation, even when factoring is difficult or impossible.

How to use this calculator

1. Enter the coefficients a, b, and c.
2. Fractions such as 1/4 are accepted.
3. Click Calculate.
4. Read the discriminant, root type, roots, and steps.

The default example is x² + x + 1/4 = 0. Its discriminant is zero, so it has one repeated real root: x = −0.5.

Quadratic formula

For ax² + bx + c = 0 with a ≠ 0:

x = (−b ± √(b² − 4ac)) ÷ 2a
Δ = b² − 4ac

The discriminant Δ determines the root type:

• Δ > 0: two distinct real roots.
• Δ = 0: one repeated real root.
• Δ < 0: two complex conjugate roots.

If a = 0, the equation is linear rather than quadratic. The calculator handles that case separately and solves bx + c = 0.

What the results mean

The roots are the values of x that make the equation true. For the default example, substituting x = −0.5 gives 0.25 − 0.5 + 0.25 = 0.

When the discriminant is negative, the roots have a real part and an imaginary part. For example, x² + 1 = 0 has the roots i and −i. The calculator writes these as 0 + 1i and 0 − 1i.

Assumptions and limitations

• The coefficients must be real numbers. Fractions are accepted.
• Results are displayed to at most ten decimal places.
• Complex roots are shown in a ± bi form but are not simplified symbolically.
• The calculator does not show a graph of the parabola.
• If a = 0 and b = 0, the equation is either always true or always false; the calculator reports that no solution exists for the false case.
• Very large coefficients may lose precision because the calculation uses double-precision floating-point numbers.

Frequently asked questions

What does the discriminant tell me?

The discriminant b² − 4ac tells you how many real roots exist. A positive value means two real roots, zero means one repeated real root, and a negative value means two complex roots.

Can I enter fractions?

Yes. Inputs such as 1/4, −3/2, and 0.25 are accepted. The calculator converts the fraction to a decimal before solving.

Why does the calculator show complex roots?

When the discriminant is negative, the square root of a negative number is imaginary. The calculator reports the real and imaginary parts instead of stopping with an error.

What happens if a is zero?

The equation is no longer quadratic. The calculator solves the remaining linear equation bx + c = 0 and labels the result as a linear equation.

Source

Reference: Calculator.net — Quadratic Formula Calculator.