What is the quadratic formula?

The quadratic formula x = (-b ± √(b²-4ac)) / (2a) finds the solutions (roots) of any quadratic equation ax² + bx + c = 0.

What does the discriminant tell us?

The discriminant (b²-4ac) tells you the nature of the roots: positive = 2 real roots, zero = 1 repeated root, negative = no real roots (2 complex roots).

What if the discriminant is negative?

If the discriminant is negative, the equation has no real solutions. The roots are complex numbers. This calculator shows the real part when the discriminant is negative.

Free Quadratic Formula Calculator

Solve quadratic equations (ax² + bx + c = 0) using the quadratic formula. Find both roots and the discriminant.

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Coefficient a

Coefficient of x²

Coefficient b

Coefficient of x

Coefficient c

Constant term

Root1

3.000000

Discriminant Val1.0000

Vertex X2.5000

Vertex Y-0.2500

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How to Use the Quadratic Formula

Formula

x = (-b ± √(b² - 4ac)) / (2a)

For the equation ax² + bx + c = 0

Discriminant (Δ = b² - 4ac)

Δ > 0: Two distinct real roots

Δ = 0: One repeated real root

Δ < 0: Two complex roots (no real solutions)

Vertex

The vertex of the parabola is at x = -b/(2a), y = f(-b/(2a)).

Example Calculation

Solve x² - 5x + 6 = 0 (a=1, b=-5, c=6).

01Discriminant = (-5)² - 4(1)(6) = 25 - 24 = 1
02Since Δ > 0, there are two distinct real roots
03x₁ = (5 + √1) / 2 = 6 / 2 = 3
04x₂ = (5 - √1) / 2 = 4 / 2 = 2
05Vertex: x = 5/2 = 2.5, y = (2.5)² - 5(2.5) + 6 = -0.25

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