1. **Problem statement:** Solve the quadratic equations by completing the square. 2. **Formula and rules:** To complete the square for an equation $x^2 + bx + c = 0$, rewrite as $x^2 + bx = -c$, then add and subtract $(\frac{b}{2})^2$ to complete the square: $$x^2 + bx + \left(\frac{b}{2}\right)^2 = -c + \left(\frac{b}{2}\right)^2$$ 3. **Solve $x^2 + 12x + 4 = 0$: ** - Rewrite: $$x^2 + 12x = -4$$ - Complete the square: add and subtract $\left(\frac{12}{2}\right)^2 = 36$: $$x^2 + 12x + 36 = -4 + 36$$ - Simplify: $$ (x + 6)^2 = 32$$ - Take square root: $$x + 6 = \pm \sqrt{32}$$ - Simplify $\sqrt{32} = \sqrt{16 \times 2} = 4\sqrt{2}$: $$x = -6 \pm 4\sqrt{2}$$ 4. **Solve $x^2 - 14x + 2 = 0$: ** - Rewrite: $$x^2 - 14x = -2$$ - Complete the square: add and subtract $\left(\frac{-14}{2}\right)^2 = 49$: $$x^2 - 14x + 49 = -2 + 49$$ - Simplify: $$ (x - 7)^2 = 47$$ - Take square root: $$x - 7 = \pm \sqrt{47}$$ - Final: $$x = 7 \pm \sqrt{47}$$ 5. **Solve $x^2 - 5x = 4$ (approximate):** - Rewrite: $$x^2 - 5x - 4 = 0$$ - Complete the square: move constant: $$x^2 - 5x = 4$$ - Add and subtract $\left(\frac{-5}{2}\right)^2 = \frac{25}{4}$: $$x^2 - 5x + \frac{25}{4} = 4 + \frac{25}{4}$$ - Simplify right side: $$ (x - \frac{5}{2})^2 = \frac{16}{4} + \frac{25}{4} = \frac{41}{4}$$ - Take square root: $$x - \frac{5}{2} = \pm \frac{\sqrt{41}}{2}$$ - Final approximate solutions: $$x = 2.5 \pm 3.2$$ - So, $$x \approx 5.7 \text{ or } -0.7$$ 6. **Solve $6x^2 - 60x + 30 = 0$ (approximate):** - Divide whole equation by 6: $$\cancel{6}x^2 - \cancel{6}10x + \cancel{6}5 = 0 \Rightarrow x^2 - 10x + 5 = 0$$ - Rewrite: $$x^2 - 10x = -5$$ - Complete the square: add and subtract $\left(\frac{-10}{2}\right)^2 = 25$: $$x^2 - 10x + 25 = -5 + 25$$ - Simplify: $$ (x - 5)^2 = 20$$ - Take square root: $$x - 5 = \pm \sqrt{20} = \pm 2\sqrt{5}$$ - Final approximate solutions: $$x = 5 \pm 4.47$$ - So, $$x \approx 9.5 \text{ or } 0.5$$ 7. **Solve $6x^2 - 48x + 18 = 0$ (approximate):** - Divide whole equation by 6: $$\cancel{6}x^2 - \cancel{6}8x + \cancel{6}3 = 0 \Rightarrow x^2 - 8x + 3 = 0$$ - Rewrite: $$x^2 - 8x = -3$$ - Complete the square: add and subtract $\left(\frac{-8}{2}\right)^2 = 16$: $$x^2 - 8x + 16 = -3 + 16$$ - Simplify: $$ (x - 4)^2 = 13$$ - Take square root: $$x - 4 = \pm \sqrt{13}$$ - Final approximate solutions: $$x = 4 \pm 3.6$$ - So, $$x \approx 7.6 \text{ or } 0.4$$ 8. **Solve $2x^2 + 10x = 1$ (approximate):** - Rewrite: $$2x^2 + 10x - 1 = 0$$ - Divide whole equation by 2: $$x^2 + 5x = \frac{1}{2}$$ - Complete the square: add and subtract $\left(\frac{5}{2}\right)^2 = \frac{25}{4}$: $$x^2 + 5x + \frac{25}{4} = \frac{1}{2} + \frac{25}{4}$$ - Simplify right side: $$ (x + \frac{5}{2})^2 = \frac{2}{4} + \frac{25}{4} = \frac{27}{4}$$ - Take square root: $$x + \frac{5}{2} = \pm \frac{\sqrt{27}}{2} = \pm \frac{3\sqrt{3}}{2}$$ - Final approximate solutions: $$x = -2.5 \pm 2.6$$ - So, $$x \approx 0.1 \text{ or } -5.1$$