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)

Derive the formula without skipping the algebra

The formula is not a separate trick. It is what completing the square produces when the coefficients stay symbolic.

ax^2 + bx + c = 0Start with a nonzero a so the equation is genuinely quadratic.
x^2 + (b/a)x = -c/aDivide every term by a, then move the constant term.
(x + b/(2a))^2 = (b^2 - 4ac)/(4a^2)Add (b/(2a))^2 to both sides and combine the right side.
x + b/(2a) = +/-sqrt(b^2 - 4ac)/(2a)Taking a square root creates both branches.
x = (-b +/- sqrt(b^2 - 4ac))/(2a)Subtract b/(2a) and combine the numerator.

If a is negative, writing the square-root fraction with denominator 2a still gives the same two roots; the plus and minus labels simply swap.

Worked example 1

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

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 example 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

Sign checks that prevent most errors

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.