Lesson 13.1: Squares of Binomials Revisited

Property

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.

Property

We can use extraction of roots to solve quadratic equations of the form

We start by isolating the squared expression, $(x - p)^2$.

Examples

• To solve $5(x - 3)^2 = 125$, first divide by 5 to get $(x - 3)^2 = 25$. Take the square root: $x - 3 = ±5$. This gives two equations: $x-3=5$ (so $x=8$) and $x-3=-5$ (so $x=-2$).
• To solve $2(x+4)^2 - 8 = 0$, add 8 and divide by 2 to get $(x+4)^2=4$. Take the square root: $x+4 = ±2$. The solutions are $x=-2$ and $x=-6$.
• In $-3(x+1)^2 = -75$, divide by -3 to get $(x+1)^2=25$. Take the square root: $x+1 = ±5$. The solutions are $x=4$ and $x=-6$.

Explanation

This method extends extraction of roots to cases where an entire expression in parentheses is squared. The strategy is the same: treat the squared parenthesis as a single block, isolate it, take the square root of both sides, and then solve for $x$.