1. **Problem Statement:** Determine the number to add to each expression to make it a perfect square trinomial and then express it as a square of a binomial. 2. **Formula Used:** To complete the square for an expression of the form $x^2 + bx$, add $\left(\frac{b}{2}\right)^2$. 3. **Explanation:** This works because $\left(x + \frac{b}{2}\right)^2 = x^2 + bx + \left(\frac{b}{2}\right)^2$. 4. **Step-by-step for each:** - For $x^2 + 4x + \_\_\_$: - $b = 4$ - Number to add: $\left(\frac{4}{2}\right)^2 = 2^2 = 4$ - Expression: $x^2 + 4x + 4 = (x + 2)^2$ - For $t^2 + 10t + \_\_\_$: - $b = 10$ - Number to add: $\left(\frac{10}{2}\right)^2 = 5^2 = 25$ - Expression: $t^2 + 10t + 25 = (t + 5)^2$ - For $r^2 - 14r + \_\_\_$: - $b = -14$ - Number to add: $\left(\frac{-14}{2}\right)^2 = (-7)^2 = 49$ - Expression: $r^2 - 14r + 49 = (r - 7)^2$ - For $p^2 + 22p + \_\_\_$: - $b = 22$ - Number to add: $\left(\frac{22}{2}\right)^2 = 11^2 = 121$ - Expression: $p^2 + 22p + 121 = (p + 11)^2$ - For $x^2 - 36x + \_\_\_$: - $b = -36$ - Number to add: $\left(\frac{-36}{2}\right)^2 = (-18)^2 = 324$ - Expression: $x^2 - 36x + 324 = (x - 18)^2$ - For $w^2 + 9w + \_\_\_$: - $b = 9$ - Number to add: $\left(\frac{9}{2}\right)^2 = \left(4.5\right)^2 = 20.25$ - Expression: $w^2 + 9w + 20.25 = (w + 4.5)^2$ - For $x^2 - 11x + \_\_\_$: - $b = -11$ - Number to add: $\left(\frac{-11}{2}\right)^2 = \left(-5.5\right)^2 = 30.25$ - Expression: $x^2 - 11x + 30.25 = (x - 5.5)^2$ - For $v^2 - 25v + \_\_\_$: - $b = -25$ - Number to add: $\left(\frac{-25}{2}\right)^2 = \left(-12.5\right)^2 = 156.25$ - Expression: $v^2 - 25v + 156.25 = (v - 12.5)^2$ - For $s^2 + 3s + \_\_\_$: - $b = 3$ - Number to add: $\left(\frac{3}{2}\right)^2 = \left(1.5\right)^2 = 2.25$ - Expression: $s^2 + 3s + 2.25 = (s + 1.5)^2$ - For $r^2 - 3r + \_\_\_$: - $b = -3$ - Number to add: $\left(\frac{-3}{2}\right)^2 = \left(-1.5\right)^2 = 2.25$ - Expression: $r^2 - 3r + 2.25 = (r - 1.5)^2$ 5. **Answers to questions:** - a. The number added is $\left(\frac{b}{2}\right)^2$ where $b$ is the coefficient of the linear term. - b. Each perfect square trinomial is expressed as $\left(x + \frac{b}{2}\right)^2$ or $\left(x - \frac{|b|}{2}\right)^2$ depending on the sign of $b$. - c. Given a square of a binomial $\left(x + m\right)^2$, it expands to $x^2 + 2mx + m^2$, so the constant term is $m^2$ which is the number added to complete the square.