1. **Problem Statement:** Determine the number to add to each binomial to make it a perfect square trinomial, then express it as a square of a binomial. 2. **Formula:** To complete the square for an expression of the form $x^2 + bx$, add $\left(\frac{b}{2}\right)^2$. 3. **Step-by-step for the first problem:** - Given: $x^2 - 4x + \_\_\_$ - Here, $b = -4$. - Calculate $\left(\frac{-4}{2}\right)^2 = (-2)^2 = 4$. - Add 4 to complete the square: $x^2 - 4x + 4$. - Express as a binomial square: $(x - 2)^2$. 4. **General method for all problems:** - For each expression $r^2 + br + \_\_\_$, add $\left(\frac{b}{2}\right)^2$. - Then express as $(r + \frac{b}{2})^2$. 5. **Examples:** 1. $x^2 - 4x + 4 = (x - 2)^2$ 2. $r^2 + 10r + 25 = (r + 5)^2$ 3. $r^2 - 14r + 49 = (r - 7)^2$ 4. $r^2 + 22r + 121 = (r + 11)^2$ 5. $x^2 - 36x + 324 = (x - 18)^2$ 6. $w^2 + 9w + 20.25 = (w + 4.5)^2$ 7. $x^2 - 11x + 30.25 = (x - 5.5)^2$ 8. $y^2 + 25y + 156.25 = (y + 12.5)^2$ 9. $s^2 + 3s + 2.25 = (s + 1.5)^2$ 10. $r^2 - 4r + 4 = (r - 2)^2$ 6. **Answer to questions:** a. The number added is $\left(\frac{b}{2}\right)^2$ because completing the square requires adding the square of half the coefficient of the linear term. b. Each perfect square trinomial is expressed as $(r + \frac{b}{2})^2$ because it factors into the square of a binomial. c. To express a quadratic as a perfect square trinomial, identify the linear coefficient $b$, compute $\left(\frac{b}{2}\right)^2$, add it, and rewrite as a binomial square. Examples: - $x^2 + 6x + 9 = (x + 3)^2$ - $y^2 - 8y + 16 = (y - 4)^2$ - $z^2 + 4z + 4 = (z + 2)^2$