1. Problem 1: Solve $5x^2 = 30x$ - Divide both sides by $5$: $x^2 = 6x$ - Bring all terms to one side: $x^2 - 6x = 0$ - Factor: $x(x - 6) = 0$ - Solutions: $x = 0$ or $x = 6$ 2. Problem 2: Solve $(2x - \frac{1}{2})^2 = \frac{9}{4}$ - Take square root on both sides: $2x - \frac{1}{2} = \pm \frac{3}{2}$ - For $+$: $2x = \frac{1}{2} + \frac{3}{2} = 2 \Rightarrow x = 1$ - For $-$: $2x = \frac{1}{2} - \frac{3}{2} = -1 \Rightarrow x = -\frac{1}{2}$ 3. Problem 3: Solve $\frac{x}{x+1} + \frac{x+1}{x} = 6$ - Find common denominator $x(x+1)$ - Rewrite: $\frac{x^2 + (x+1)^2}{x(x+1)} = 6$ - Numerator: $x^2 + x^2 + 2x + 1 = 2x^2 + 2x + 1$ - Equation: $2x^2 + 2x + 1 = 6x(x+1) = 6x^2 + 6x$ - Rearrange: $2x^2 + 2x + 1 - 6x^2 - 6x = 0 \Rightarrow -4x^2 - 4x + 1=0$ - Multiply by $-1$: $4x^2 + 4x -1=0$ - Solve quadratic: $x = \frac{-4 \pm \sqrt{16 + 16}}{8} = \frac{-4 \pm \sqrt{32}}{8} = \frac{-4 \pm 4\sqrt{2}}{8} = \frac{-1 \pm \sqrt{2}}{2}$ 4. Problem 4: Solve $4 - 32x = 17x^2$ - Rearrange: $17x^2 + 32x - 4 = 0$ - Use quadratic formula: $x = \frac{-32 \pm \sqrt{32^2 - 4 \cdot 17 \cdot (-4)}}{2 \cdot 17} = \frac{-32 \pm \sqrt{1024 + 272}}{34} = \frac{-32 \pm \sqrt{1296}}{34} = \frac{-32 \pm 36}{34}$ - Solutions: $x = \frac{4}{34} = \frac{2}{17}$ or $x = \frac{-68}{34} = -2$ 5. Problem 5: Solve $4x^2 -14 = 13x$ - Rearrange: $4x^2 - 13x - 14 = 0$ - Quadratic formula: $x = \frac{13 \pm \sqrt{169 + 224}}{8} = \frac{13 \pm \sqrt{393}}{8}$ (irrational roots) 6. Problem 6: Solve $\sqrt{3x} + 18 = x$ - Rearrange: $\sqrt{3x} = x - 18$ - Square both sides: $3x = (x - 18)^2 = x^2 - 36x + 324$ - Rearrange: $x^2 - 39x + 324 = 0$ - Quadratic formula: $x = \frac{39 \pm \sqrt{1521 - 1296}}{2} = \frac{39 \pm \sqrt{225}}{2} = \frac{39 \pm 15}{2}$ - Solutions: $x=27$ or $x=12$ - Check for extraneous solutions (plug in): Both valid 7. Problem 7: Solve $3x^{-2} + 5 = 8x^{-1}$ - Substitute $y = x^{-1}$: $3y^2 + 5 = 8y$ - Rearrange: $3y^2 - 8y + 5 = 0$ - Quadratic formula: $y= \frac{8 \pm \sqrt{64 - 60}}{6} = \frac{8 \pm 2}{6}$ - Solutions: $y=\frac{10}{6} = \frac{5}{3}$ or $y=1$ - Recall $x^{-1} = y$ so $x= \frac{3}{5}$ or $x=1$ 8. Problem 8: Solve $3x^2 + 7x = 0$ - Factor: $x(3x + 7) = 0$ - Solutions: $x=0$ or $x=-\frac{7}{3}$ 9. Problem 9: Given $\omega = \frac{-1 + \sqrt{-3}}{2}$ - $\omega$ is the complex cube root of unity, satisfying $\omega^3 = 1$ and $1 + \omega + \omega^2 = 0$ 10. Problem 10: Compute $\omega^{37} + \omega^{38}$ - Use $\omega^3 = 1$, reduce powers mod 3 - $37 \equiv 1 \mod 3$, $38 \equiv 2 \mod 3$ - So $\omega^{37} + \omega^{38} = \omega^1 + \omega^2 = \omega + \omega^2$ - From roots unity: $1 + \omega + \omega^2 = 0 \Rightarrow \omega + \omega^2 = -1$ 11. Problem 11: Factor $x^3 - y^3$ - Using $\omega$ roots: $x^3 - y^3 = (x - y)(x - \omega y)(x - \omega^2 y)$ 12. Problem 12: Solve $16x^2 - 24x + 9=0$ - Compute discriminant: $\Delta = (-24)^2 - 4 \cdot 16 \cdot 9 = 576 - 576 = 0$ - Single root: $x = \frac{24}{2 \cdot 16} = \frac{24}{32} = \frac{3}{4}$ Final answers: - Problem 1: $x=0,6$ - Problem 2: $x=1, -\frac{1}{2}$ - Problem 3: $x= \frac{-1+\sqrt{2}}{2}, \frac{-1 - \sqrt{2}}{2}$ - Problem 4: $x=\frac{2}{17}, -2$ - Problem 5: $x= \frac{13 \pm \sqrt{393}}{8}$ - Problem 6: $x=27,12$ - Problem 7: $x=\frac{3}{5}, 1$ - Problem 8: $x=0, -\frac{7}{3}$ - Problem 9: Definition of $\omega$ - Problem 10: $\omega^{37}+\omega^{38} = -1$ - Problem 11: Factorization as given - Problem 12: $x=\frac{3}{4}$