1. **Problem:** Solve the quadratic equation $-3x^2 - 27 = 0$. 2. **Step 1:** Add 27 to both sides: $$-3x^2 - 27 + 27 = 0 + 27 \implies -3x^2 = 27$$ 3. **Step 2:** Divide both sides by $-3$: $$\cancel{-3}x^2 = \frac{27}{\cancel{-3}} \implies x^2 = -9$$ 4. **Step 3:** Take the square root of both sides: $$x = \pm \sqrt{-9} = \pm 3i$$ 5. **Answer:** $x = \pm 3i$ (complex roots). --- 1. **Problem:** Solve $x^2 + 4x + 5 = 0$. 2. **Step 1:** Use quadratic formula $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ with $a=1$, $b=4$, $c=5$. 3. **Step 2:** Calculate discriminant: $$b^2 - 4ac = 4^2 - 4(1)(5) = 16 - 20 = -4$$ 4. **Step 3:** Substitute into formula: $$x = \frac{-4 \pm \sqrt{-4}}{2} = \frac{-4 \pm 2i}{2}$$ 5. **Step 4:** Simplify: $$x = -2 \pm i$$ 6. **Answer:** $x = -2 \pm i$ (complex roots). --- 1. **Problem:** Solve $5x^2 - 14x - 3 = 0$. 2. **Step 1:** Use quadratic formula with $a=5$, $b=-14$, $c=-3$. 3. **Step 2:** Calculate discriminant: $$(-14)^2 - 4(5)(-3) = 196 + 60 = 256$$ 4. **Step 3:** Substitute: $$x = \frac{14 \pm \sqrt{256}}{2 \times 5} = \frac{14 \pm 16}{10}$$ 5. **Step 4:** Calculate roots: $$x_1 = \frac{14 + 16}{10} = \frac{30}{10} = 3$$ $$x_2 = \frac{14 - 16}{10} = \frac{-2}{10} = -\frac{1}{5}$$ 6. **Answer:** $x = 3$ or $x = -\frac{1}{5}$. --- 1. **Problem:** Solve $3x^2 - 22x + 7 = 0$. 2. **Step 1:** Use quadratic formula with $a=3$, $b=-22$, $c=7$. 3. **Step 2:** Calculate discriminant: $$(-22)^2 - 4(3)(7) = 484 - 84 = 400$$ 4. **Step 3:** Substitute: $$x = \frac{22 \pm \sqrt{400}}{2 \times 3} = \frac{22 \pm 20}{6}$$ 5. **Step 4:** Calculate roots: $$x_1 = \frac{22 + 20}{6} = \frac{42}{6} = 7$$ $$x_2 = \frac{22 - 20}{6} = \frac{2}{6} = \frac{1}{3}$$ 6. **Answer:** $x = 7$ or $x = \frac{1}{3}$. --- 1. **Problem:** Solve $\log_2(x + 2) + \log_2(x) = 3$. 2. **Step 1:** Use log property: $\log_b A + \log_b B = \log_b (AB)$. 3. **Step 2:** Combine logs: $$\log_2(x(x+2)) = 3$$ 4. **Step 3:** Convert to exponential form: $$2^3 = x(x+2)$$ $$8 = x^2 + 2x$$ 5. **Step 4:** Rearrange: $$x^2 + 2x - 8 = 0$$ 6. **Step 5:** Factor: $$(x + 4)(x - 2) = 0$$ 7. **Step 6:** Solve: $$x = -4 \text{ or } x = 2$$ 8. **Step 7:** Check domain (arguments of logs must be positive): $$x > 0, x+2 > 0$$ 9. **Step 8:** $x = -4$ invalid, $x = 2$ valid. 10. **Answer:** $x = 2$. --- 1. **Problem:** Solve $\log_8 x + \log_8 (x + 6) = \log_8 (5x + 12)$. 2. **Step 1:** Use log property: $$\log_8 (x(x+6)) = \log_8 (5x + 12)$$ 3. **Step 2:** Equate arguments: $$x(x+6) = 5x + 12$$ 4. **Step 3:** Expand and rearrange: $$x^2 + 6x - 5x - 12 = 0$$ $$x^2 + x - 12 = 0$$ 5. **Step 4:** Factor: $$(x + 4)(x - 3) = 0$$ 6. **Step 5:** Solve: $$x = -4 \text{ or } x = 3$$ 7. **Step 6:** Check domain: $$x > 0, x+6 > 0$$ 8. **Step 7:** $x = -4$ invalid, $x = 3$ valid. 9. **Answer:** $x = 3$. --- 1. **Problem:** Solve $2x^2 + 7x + 3 = 0$. 2. **Step 1:** Use quadratic formula with $a=2$, $b=7$, $c=3$. 3. **Step 2:** Calculate discriminant: $$7^2 - 4(2)(3) = 49 - 24 = 25$$ 4. **Step 3:** Substitute: $$x = \frac{-7 \pm 5}{2 \times 2} = \frac{-7 \pm 5}{4}$$ 5. **Step 4:** Calculate roots: $$x_1 = \frac{-7 + 5}{4} = \frac{-2}{4} = -\frac{1}{2}$$ $$x_2 = \frac{-7 - 5}{4} = \frac{-12}{4} = -3$$ 6. **Answer:** $x = -\frac{1}{2}$ or $x = -3$. --- 1. **Problem:** Solve $2(x - 3)^2 - 5 = 7$. 2. **Step 1:** Add 5 to both sides: $$2(x - 3)^2 = 12$$ 3. **Step 2:** Divide both sides by 2: $$\cancel{2}(x - 3)^2 = \frac{12}{\cancel{2}} \implies (x - 3)^2 = 6$$ 4. **Step 3:** Take square root: $$x - 3 = \pm \sqrt{6}$$ 5. **Step 4:** Solve for $x$: $$x = 3 \pm \sqrt{6}$$ 6. **Answer:** $x = 3 + \sqrt{6}$ or $x = 3 - \sqrt{6}$. --- 1. **Problem:** Solve $2x^2 - 6x = 0$. 2. **Step 1:** Factor out $2x$: $$2x(x - 3) = 0$$ 3. **Step 2:** Set each factor to zero: $$2x = 0 \implies x = 0$$ $$x - 3 = 0 \implies x = 3$$ 4. **Answer:** $x = 0$ or $x = 3$. --- 1. **Problem:** Solve $(x - 1)^2 - 2 = -3$. 2. **Step 1:** Add 2 to both sides: $$(x - 1)^2 = -3 + 2 = -1$$ 3. **Step 2:** Take square root: $$x - 1 = \pm \sqrt{-1} = \pm i$$ 4. **Step 3:** Solve for $x$: $$x = 1 \pm i$$ 5. **Answer:** $x = 1 \pm i$. --- 1. **Problem:** Solve $5x^2 - 16x + 10 = 0$. 2. **Step 1:** Use quadratic formula with $a=5$, $b=-16$, $c=10$. 3. **Step 2:** Calculate discriminant: $$(-16)^2 - 4(5)(10) = 256 - 200 = 56$$ 4. **Step 3:** Substitute: $$x = \frac{16 \pm \sqrt{56}}{10} = \frac{16 \pm 2\sqrt{14}}{10}$$ 5. **Step 4:** Simplify: $$x = \frac{8 \pm \sqrt{14}}{5}$$ 6. **Answer:** $x = \frac{8 + \sqrt{14}}{5}$ or $x = \frac{8 - \sqrt{14}}{5}$. --- 1. **Problem:** Solve $x^2 - 2x + 10 = 0$. 2. **Step 1:** Use quadratic formula with $a=1$, $b=-2$, $c=10$. 3. **Step 2:** Calculate discriminant: $$(-2)^2 - 4(1)(10) = 4 - 40 = -36$$ 4. **Step 3:** Substitute: $$x = \frac{2 \pm \sqrt{-36}}{2} = \frac{2 \pm 6i}{2}$$ 5. **Step 4:** Simplify: $$x = 1 \pm 3i$$ 6. **Answer:** $x = 1 \pm 3i$ (complex roots).