Quadratic Formula & The Discriminant Cheat Sheet

Key Facts

• A quadratic equation in standard form is $ax^2 + bx + c = 0$, where $a \ne 0$.
• The quadratic formula is $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$.
• The discriminant is $D = b^2 - 4ac$ and it appears under the square root in the quadratic formula.
• If $D > 0$, the equation has two distinct real solutions.
• If $D = 0$, the equation has one real repeated solution, $x = \frac{-b}{2a}$.
• If $D < 0$, the equation has two complex solutions because the square root of a negative number is not real.
• The axis of symmetry of the graph $y = ax^2 + bx + c$ is $x = \frac{-b}{2a}$.
• The solutions of $ax^2 + bx + c = 0$ are the $x$-intercepts of the graph $y = ax^2 + bx + c$ when the solutions are real.

Vocabulary

Quadratic equation

A polynomial equation of degree $2$ that can be written as $ax^2 + bx + c = 0$ with $a \ne 0$.

Quadratic formula

The formula $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ used to solve any quadratic equation in standard form.

Discriminant

The expression $D = b^2 - 4ac$ that tells the number and type of solutions of a quadratic equation.

Standard form

The form $ax^2 + bx + c = 0$, where the terms are arranged by descending powers of $x$ and one side equals $0$.

Repeated solution

A single real solution that occurs when $D = 0$ and the parabola touches the $x$-axis at exactly one point.

Complex solution

A solution involving the imaginary unit $i$, which occurs when the discriminant is negative.