1. Problem: Simplify the following complex quotients and express each result in the form $a+ib$. 2. (a) Compute $\frac{20}{3+i}$. Multiply numerator and denominator by the conjugate $3 - i$. $$\frac{20}{3+i}=\frac{20(3 - i)}{(3+i)(3 - i)}=\frac{20(3 - i)}{10}=6 - 2i$$ 3. (b) Compute $\frac{4}{1+i}$. Multiply top and bottom by $1 - i$. $$\frac{4}{1+i}=\frac{4(1 - i)}{(1+i)(1 - i)}=\frac{4(1 - i)}{2}=2(1 - i)=2 - 2i$$ 4. (c) Compute $\frac{21}{1 - 2i}$. Multiply by the conjugate $1 + 2i$. $$\frac{21}{1 - 2i}=\frac{21(1 + 2i)}{(1 - 2i)(1 + 2i)}=\frac{21(1 + 2i)}{1+4}=\frac{21}{5}+\frac{42}{5}i$$ 5. (d) Compute $\frac{1}{3 - i}$. Multiply by $3 + i$. $$\frac{1}{3 - i}=\frac{3 + i}{(3 - i)(3 + i)}=\frac{3 + i}{10}=\frac{3}{10}+\frac{1}{10}i$$ 6. (e1) Compute $\frac{5i}{1 + 2i}$. Multiply by the conjugate $1 - 2i$. $$\frac{5i}{1 + 2i}=\frac{5i(1 - 2i)}{1+4}=\frac{5i -10i^2}{5}=\frac{5i+10}{5}=2+i$$ 7. (e2) Compute $\frac{3 + 2i}{3 - 2i}$. Multiply by $3 + 2i$. $$\frac{3 + 2i}{3 - 2i}=\frac{(3 + 2i)^2}{(3 - 2i)(3 + 2i)}=\frac{9+12i-4}{9+4}=\frac{5+12i}{13}=\frac{5}{13}+\frac{12}{13}i$$ 8. (f1) Compute $\frac{5}{4 - 3i}$. Multiply by the conjugate $4 + 3i$. $$\frac{5}{4 - 3i}=\frac{5(4 + 3i)}{16+9}=\frac{5(4 + 3i)}{25}=\frac{4}{5}+\frac{3}{5}i$$ 9. (f2) Compute $\frac{4i - 3}{2 + 3i}$. Multiply numerator and denominator by $2 - 3i$ and expand. $$\frac{4i - 3}{2 + 3i}=\frac{(4i - 3)(2 - 3i)}{4+9}=\frac{6+17i}{13}=\frac{6}{13}+\frac{17}{13}i$$ 10. (g) Compute $\frac{2 + 3i}{1 - i}$. Multiply by $1 + i$. $$\frac{2 + 3i}{1 - i}=\frac{(2 + 3i)(1 + i)}{2}=\frac{-1+5i}{2}=-\frac{1}{2}+\frac{5}{2}i$$ 11. (h) Compute $\frac{3 - i}{1 + 2i}$. Multiply by $1 - 2i$. $$\frac{3 - i}{1 + 2i}=\frac{(3 - i)(1 - 2i)}{5}=\frac{1 -7i}{5}=\frac{1}{5}-\frac{7}{5}i$$ 12. Problem 5: Simplify $\frac{a+ib}{b - ai}$ where $a,b$ are real. Multiply numerator and denominator by the conjugate $b + ai$ and expand. $$\frac{a+ib}{b - ai}=\frac{(a+ib)(b + ai)}{b^2+a^2}=\frac{i(a^2+b^2)}{a^2+b^2}=i$$ 13. Problem 6(a): Solve $x^2+25=0$. This gives $x^2=-25$ so $x=\pm 5i$. 14. Problem 6(b): Solve $2x^2+32=0$. Divide by 2 to get $x^2=-16$ so $x=\pm 4i$. 15. Problem 6(c): Solve $4x^2+9=0$. This gives $x^2=-9/4$ so $x=\pm \frac{3}{2}i$. 16. Problem 6(d): Solve $x^2+2x+5=0$. Use the quadratic formula to get $x=\frac{-2\pm\sqrt{4-20}}{2}= -1\pm 2i$. 17. Problem 6(e): Solve $x^2-4x+5=0$. Quadratic formula gives $x=\frac{4\pm\sqrt{16-20}}{2}=2\pm i$. 18. Problem 6(f): Solve $2x^2+x+1=0$. Quadratic formula gives $x=\frac{-1\pm\sqrt{1-8}}{4}=\frac{-1\pm i\sqrt{7}}{4}$