1. **Problem:** Approximate $\sqrt{66}$ without a calculator and justify with inequalities. 2. **Problem:** Approximate $\sqrt{46}$ without a calculator and justify with inequalities. 3. **Problem:** Simplify $\sqrt{200}$. 4. **Problem:** Simplify $6\sqrt{48} \cdot \sqrt{30}$. 5. **Problem:** Simplify $-3\sqrt{45} + 2\sqrt{5}$. 6. **Problem:** Solve $53 = 8 + 9m^2$ for $m$. 7. **Problem:** Solve $\frac{(x+3)^2}{3} + 15 = 27$ for $x$. 8. **Problem:** Solve $(x + 6)^2 - 12 = -2$ and round to nearest hundredth. 9. **Problem:** Solve $x^2 + 9x = 3x + 7$ using the quadratic formula. 10. **Problem:** Solve $8x^2 + 8x - 3 = 12x^2$ using the quadratic formula. 11. **Problem:** Solve $5x^2 + 3x + 1 = -x^2 + 7x$ using the quadratic formula. --- ### Step-by-step solutions: 1. Approximate $\sqrt{66}$: - Find perfect squares near 66: $8^2=64$ and $9^2=81$. - Since $64 < 66 < 81$, then $8 < \sqrt{66} < 9$. - Approximate: $\sqrt{66} \approx 8.1$ (since 66 is closer to 64 than 81). 2. Approximate $\sqrt{46}$: - Perfect squares near 46: $6^2=36$ and $7^2=49$. - Since $36 < 46 < 49$, then $6 < \sqrt{46} < 7$. - Approximate: $\sqrt{46} \approx 6.8$ (closer to 7). 3. Simplify $\sqrt{200}$: - Factor 200: $200 = 100 \times 2$. - Use $\sqrt{ab} = \sqrt{a} \times \sqrt{b}$: $$\sqrt{200} = \sqrt{100 \times 2} = \sqrt{100} \times \sqrt{2} = 10\sqrt{2}$$ 4. Simplify $6\sqrt{48} \cdot \sqrt{30}$: - Combine radicals: $6 \times \sqrt{48 \times 30} = 6 \sqrt{1440}$. - Factor 1440: $1440 = 144 \times 10$. - Simplify: $$6 \sqrt{1440} = 6 \times \sqrt{144 \times 10} = 6 \times \sqrt{144} \times \sqrt{10} = 6 \times 12 \times \sqrt{10} = 72\sqrt{10}$$ 5. Simplify $-3\sqrt{45} + 2\sqrt{5}$: - Simplify $\sqrt{45}$: $45 = 9 \times 5$, so $\sqrt{45} = 3\sqrt{5}$. - Substitute: $$-3\sqrt{45} + 2\sqrt{5} = -3 \times 3\sqrt{5} + 2\sqrt{5} = -9\sqrt{5} + 2\sqrt{5} = (-9 + 2)\sqrt{5} = -7\sqrt{5}$$ 6. Solve $53 = 8 + 9m^2$: - Subtract 8 from both sides: $$53 - 8 = 9m^2 \Rightarrow 45 = 9m^2$$ - Divide both sides by 9: $$\cancel{9}m^2 = \frac{45}{\cancel{9}} \Rightarrow m^2 = 5$$ - Take square root: $$m = \pm \sqrt{5}$$ 7. Solve $\frac{(x+3)^2}{3} + 15 = 27$: - Subtract 15: $$\frac{(x+3)^2}{3} = 12$$ - Multiply both sides by 3: $$\cancel{3} \times \frac{(x+3)^2}{\cancel{3}} = 12 \times 3 \Rightarrow (x+3)^2 = 36$$ - Take square root: $$x+3 = \pm 6$$ - Solve for $x$: $$x = -3 \pm 6$$ - Solutions: $$x = 3 \text{ or } x = -9$$ 8. Solve $(x + 6)^2 - 12 = -2$ and round to nearest hundredth: - Add 12 to both sides: $$(x + 6)^2 = 10$$ - Take square root: $$x + 6 = \pm \sqrt{10}$$ - Approximate $\sqrt{10} \approx 3.1623$. - Solve for $x$: $$x = -6 \pm 3.1623$$ - Solutions: $$x_1 = -6 + 3.1623 = -2.84$$ $$x_2 = -6 - 3.1623 = -9.16$$ 9. Solve $x^2 + 9x = 3x + 7$ using quadratic formula: - Rearrange: $$x^2 + 9x - 3x - 7 = 0 \Rightarrow x^2 + 6x - 7 = 0$$ - Quadratic formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ with $a=1$, $b=6$, $c=-7$. - Calculate discriminant: $$\Delta = 6^2 - 4 \times 1 \times (-7) = 36 + 28 = 64$$ - Calculate roots: $$x = \frac{-6 \pm \sqrt{64}}{2} = \frac{-6 \pm 8}{2}$$ - Solutions: $$x_1 = \frac{-6 + 8}{2} = 1$$ $$x_2 = \frac{-6 - 8}{2} = -7$$ 10. Solve $8x^2 + 8x - 3 = 12x^2$ using quadratic formula: - Rearrange: $$8x^2 + 8x - 3 - 12x^2 = 0 \Rightarrow -4x^2 + 8x - 3 = 0$$ - Multiply both sides by $-1$ for easier handling: $$4x^2 - 8x + 3 = 0$$ - Quadratic formula with $a=4$, $b=-8$, $c=3$. - Discriminant: $$\Delta = (-8)^2 - 4 \times 4 \times 3 = 64 - 48 = 16$$ - Roots: $$x = \frac{8 \pm \sqrt{16}}{2 \times 4} = \frac{8 \pm 4}{8}$$ - Solutions: $$x_1 = \frac{8 + 4}{8} = \frac{12}{8} = 1.5$$ $$x_2 = \frac{8 - 4}{8} = \frac{4}{8} = 0.5$$ 11. Solve $5x^2 + 3x + 1 = -x^2 + 7x$ using quadratic formula: - Rearrange: $$5x^2 + 3x + 1 + x^2 - 7x = 0 \Rightarrow 6x^2 - 4x + 1 = 0$$ - Quadratic formula with $a=6$, $b=-4$, $c=1$. - Discriminant: $$\Delta = (-4)^2 - 4 \times 6 \times 1 = 16 - 24 = -8$$ - Since $\Delta < 0$, solutions are complex: $$x = \frac{4 \pm \sqrt{-8}}{12} = \frac{4 \pm 2i\sqrt{2}}{12} = \frac{4}{12} \pm \frac{2i\sqrt{2}}{12} = \frac{1}{3} \pm \frac{i\sqrt{2}}{6}$$ --- **Final answers:** 1. $\sqrt{66} \approx 8.1$ with $8 < \sqrt{66} < 9$ 2. $\sqrt{46} \approx 6.8$ with $6 < \sqrt{46} < 7$ 3. $10\sqrt{2}$ 4. $72\sqrt{10}$ 5. $-7\sqrt{5}$ 6. $m = \pm \sqrt{5}$ 7. $x = 3$ or $x = -9$ 8. $x \approx -2.84$ or $x \approx -9.16$ 9. $x = 1$ or $x = -7$ 10. $x = 1.5$ or $x = 0.5$ 11. $x = \frac{1}{3} \pm \frac{i\sqrt{2}}{6}$ (complex solutions)