Completing the Square Reference Cheat Sheet

Key Facts

• A perfect square trinomial has the form $x^2 + 2px + p^2 = (x + p)^2$.
• To complete the square for $x^2 + bx$, add $\left(\frac{b}{2}\right)^2$ to make $x^2 + bx + \left(\frac{b}{2}\right)^2 = \left(x + \frac{b}{2}\right)^2$.
• When solving $x^2 + bx + c = 0$, first move the constant to get $x^2 + bx = -c$.
• If you add $\left(\frac{b}{2}\right)^2$ to one side of an equation, you must add $\left(\frac{b}{2}\right)^2$ to the other side too.
• To solve after completing the square, use the square root property: if $(x + p)^2 = q$, then $x + p = \pm \sqrt{q}$.
• For $ax^2 + bx + c$ with $a \neq 1$, factor $a$ from the $x^2$ and $x$ terms before completing the square.
• Vertex form is $y = a(x - h)^2 + k$, and the vertex of the parabola is $(h, k)$.
• Completing the square leads to the quadratic formula $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$.

Vocabulary

Completing the Square

A method of rewriting a quadratic expression by creating a perfect square trinomial.

Perfect Square Trinomial

A trinomial that can be factored as the square of a binomial, such as $x^2 + 6x + 9 = (x + 3)^2$.

Vertex Form

A quadratic function written as $y = a(x - h)^2 + k$, where $(h, k)$ is the vertex.

Square Root Property

The rule that if $u^2 = q$, then $u = \pm \sqrt{q}$.

Coefficient

A number multiplying a variable, such as $b$ in $bx$ or $a$ in $ax^2$.

Vertex

The highest or lowest point of a parabola, written as $(h, k)$ in vertex form.