Quadratic Formula

The quadratic formula solves every quadratic equation and reveals how its roots depend on the shape of its parabola.

The vertex anchors a parabola’s position and symmetry. A U-shaped curve is mirrored across a dashed vertical axis. Its highlighted vertex is the turning point, so vertex form and the axis of symmetry can be read from the same picture.

The quadratic formula solves any equation of the form $ax^2+bx+c=0$, where $a\ne0$.

The idea in one minute

A quadratic equation asks where a parabola reaches height zero. Those locations are its roots, zeros, or solutions—three views of the same $x$-values.

Some quadratics factor neatly. Many do not. The quadratic formula works in both cases:

x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}

First write the equation in standard form:

$$ ax^2+bx+c=0. $$

Then $a$, $b$, and $c$ are the coefficients, including their signs.

A dependable five-step method

Rearrange the equation so one side is zero.
Record $a$, $b$, and $c$ with their signs.
Calculate the discriminant $D=b^2-4ac$.
Substitute into $x=(-b\pm\sqrt D)/(2a)$.
Simplify both branches and check when practical.

Writing the coefficient list before substituting prevents most avoidable errors.

Worked example: two rational roots

Why the formula works

Begin with $ax^2+bx+c=0$ and divide by $a$:

x^2+\frac ba x=-\frac ca.

Complete the square by adding $(b/2a)^2$ to both sides:

\left(x+\frac b{2a}\right)^2 =\frac{b^2-4ac}{4a^2}.

Take both square roots and isolate $x$:

x+\frac b{2a}=\pm\frac{\sqrt{b^2-4ac}}{2a},

x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}.

The quadratic formula is Completing the Square applied once to every possible quadratic.

Common mistakes

Choosing a solving method

Use factoring when the factors are visible.
Use square roots when the equation looks like $(x-h)^2=k$.
Use completing the square when vertex form matters.
Use the quadratic formula when you want a method that always works.