Algebra

How the Quadratic Formula Works

The quadratic formula solves every equation of the form ax^2 + bx + c = 0 with a not equal to zero. It is the result of completing the square for general coefficients.

Reviewed 2026-07-26

The formula and its conditions

For ax^2 + bx + c = 0, the coefficient a must be nonzero; otherwise the equation is linear. The plus-or-minus sign represents two possible values that are symmetric around -b/(2a).

The expression under the square root, b^2 - 4ac, determines whether those two values are distinct real roots, one repeated real root, or a complex-conjugate pair.

Formulax = (-b +/- sqrt(b^2 - 4ac)) / (2a)

Deriving it by completing the square

Divide the equation by a, move the constant term to the other side, and add the square of half the x coefficient to both sides. The left side becomes a perfect square.

Taking square roots introduces both signs. Isolating x and combining the fractions produces the standard formula.

x^2 + (b/a)x = -c/a
x^2 + (b/a)x + (b/2a)^2 = -c/a + (b/2a)^2
(x + b/2a)^2 = (b^2 - 4ac)/(4a^2)
x = -b/(2a) +/- sqrt(b^2 - 4ac)/(2a)

Reading the geometry

The graph y = ax^2 + bx + c is a parabola. Its axis of symmetry is x = -b/(2a), the midpoint of the two roots when real roots exist.

A root is an x-coordinate where the parabola meets the x-axis. Complex roots arise when the real parabola never reaches that axis.

Worked examples

Solve 2x^2 - 3x - 2 = 0

Use a = 2, b = -3, and c = -2.

D = (-3)^2 - 4(2)(-2) = 25
x = (3 +/- 5)/4
x = 8/4 or x = -2/4

Result: x = 2 or x = -1/2

Solve x^2 + 4x + 8 = 0

The discriminant is negative.

D = 16 - 32 = -16
sqrt(-16) = 4i
x = (-4 +/- 4i)/2

Result: x = -2 +/- 2i

Where this idea is used

Finding where a projectile model reaches a chosen height.
Solving area and optimization models that produce quadratic equations.
Checking roots obtained by factoring or graphing.

Common mistakes

Using b without its sign; if b = -3, then -b = 3.
Dividing only the square-root term by 2a instead of the entire numerator.
Forgetting that a = 0 makes the quadratic formula inapplicable.

Questions about the quadratic formula

Does the quadratic formula always give two different answers?

No. A zero discriminant gives one repeated root, and some equations have the same root listed twice.

Can it solve equations that factor easily?

Yes. Factoring may be faster, but both methods produce the same roots.

Why does a negative discriminant produce i?

The square root of a negative real number is represented using the imaginary unit, so the roots lie in the complex number system.