1. **Problem 1:** Write $x^2 + 8x + 5$ in the form $(x + a)^2 + b$ where $a,b$ are integers. 2. **Formula:** To complete the square for $x^2 + bx + c$, use: $$x^2 + bx + c = (x + \frac{b}{2})^2 + c - \left(\frac{b}{2}\right)^2$$ 3. **Step:** For $x^2 + 8x + 5$, $b=8$, so $a=\frac{8}{2}=4$. 4. Calculate $b$: $$b = 5 - 4^2 = 5 - 16 = -11$$ 5. **Answer:** $$x^2 + 8x + 5 = (x + 4)^2 - 11$$ --- 1. **Problem 2:** Write $x^2 - 10x - 9$ in the form $(x + a)^2 + b$. 2. Here, $b = -10$, so $a = \frac{-10}{2} = -5$. 3. Calculate $b$: $$b = -9 - (-5)^2 = -9 - 25 = -34$$ 4. **Answer:** $$x^2 - 10x - 9 = (x - 5)^2 - 34$$ --- 1. **Problem 3:** Express $x^2 - 5x + 1$ in completed square form. 2. $b = -5$, so $a = \frac{-5}{2} = -\frac{5}{2}$ (not an integer, but we keep it for calculation). 3. Calculate $b$: $$b = 1 - \left(-\frac{5}{2}\right)^2 = 1 - \frac{25}{4} = \frac{4}{4} - \frac{25}{4} = -\frac{21}{4}$$ 4. **Answer:** $$x^2 - 5x + 1 = \left(x - \frac{5}{2}\right)^2 - \frac{21}{4}$$ (Note: $a,b$ are not integers here.) --- 1. **Problem 4:** Write $w^2 + 9w - 3$ in completed square form. 2. $b=9$, so $a=\frac{9}{2} = 4.5$ (not integer). 3. Calculate $b$: $$b = -3 - (4.5)^2 = -3 - 20.25 = -23.25$$ 4. **Answer:** $$w^2 + 9w - 3 = (w + 4.5)^2 - 23.25$$ --- 1. **Problem 5:** Express $2x^2 + 12x + 6$ in the form $a(x + b)^2 + c$ with integers $a,b,c$. 2. Factor out 2: $$2x^2 + 12x + 6 = 2(x^2 + 6x) + 6$$ 3. Complete the square inside parentheses: $$x^2 + 6x = (x + 3)^2 - 9$$ 4. Substitute back: $$2[(x + 3)^2 - 9] + 6 = 2(x + 3)^2 - 18 + 6 = 2(x + 3)^2 - 12$$ 5. **Answer:** $$a=2, b=3, c=-12$$ --- 1. **Problem 6:** Express $2x^2 + 16x + 7$ in the form $a(x + b)^2 + c$. 2. Factor out 2: $$2x^2 + 16x + 7 = 2(x^2 + 8x) + 7$$ 3. Complete the square: $$x^2 + 8x = (x + 4)^2 - 16$$ 4. Substitute back: $$2[(x + 4)^2 - 16] + 7 = 2(x + 4)^2 - 32 + 7 = 2(x + 4)^2 - 25$$ 5. **Answer:** $$a=2, b=4, c=-25$$ --- 1. **Problem 7:** Solve $(x - 7)^2 - 10 = 0$ for $x$ in the form $x = a \pm \sqrt{b}$. 2. Add 10 to both sides: $$(x - 7)^2 = 10$$ 3. Take square root: $$x - 7 = \pm \sqrt{10}$$ 4. Solve for $x$: $$x = 7 \pm \sqrt{10}$$ 5. **Answer:** $$x = 7 \pm \sqrt{10}$$