Lesson 13.2: Completing the Square

Property

To make a perfect square trinomial from an expression like $x^2 + bx$, we use the binomial squares pattern. The goal is to find a number to add that completes the pattern $a^2 + 2ab + b^2 = (a+b)^2$.

To complete the square of $x^2 + bx$:
Step 1. Identify $b$, the coefficient of $x$.
Step 2. Find $(\frac{1}{2}b)^2$, the number to complete the square.
Step 3. Add the $(\frac{1}{2}b)^2$ to $x^2 + bx$. The result is $x^2 + bx + (\frac{1}{2}b)^2$, which factors to $(x + \frac{b}{2})^2$.

Examples

• To complete the square for $x^2 + 12x$, we find $(\frac{1}{2} \cdot 12)^2 = 6^2 = 36$. The perfect square trinomial is $x^2 + 12x + 36$, which factors to $(x+6)^2$.

Property

To solve an equation by extraction of roots, first isolate the squared expression. Then, take the square root of both sides, remembering to include both the positive and negative roots ($\pm$). Finally, solve the resulting linear equations.

Examples

• Solve $(x+3)^2 = 49$. Take the square root of both sides: $x+3 = \pm\sqrt{49} = \pm 7$. This gives two equations: $x+3=7$ (so $x=4$) and $x+3=-7$ (so $x=-10$).

• Solve $3(y-1)^2 = 75$. First, isolate the squared part by dividing by 3: $(y-1)^2 = 25$. Take the square root: $y-1 = \pm 5$. The two solutions are $y = 1+5=6$ and $y=1-5=-4$.