#10 \(\frac{15}{2i}\)
• Multiply by \(\frac{i}{i}\): \(\frac{15i}{2i^2}\)
• Denominator: \(2(-1) = -2\)
• Result: \(\frac{15i}{-2} = -\frac{15}{2}i\)
Answer: \(-\frac{15}{2}i\)
#11 \(\frac{5 + i}{3 - 2i}\)
• Conjugate: \(3 + 2i\)
• Numerator: \((5+i)(3+2i) = 13 + 13i\)
• Denom: \(3^2 - (2i)^2 = 9 + 4 = 13\)
• Divide: \(\frac{13+13i}{13} = 1 + i\)
Answer: \(1 + i\)
#12 \(\frac{7 - 4i}{1 + 3i}\)
• Conjugate: \(1 - 3i\)
• Numerator: \((7-4i)(1-3i) = -5 - 25i\)
• Denom: \(1^2 - (3i)^2 = 1 + 9 = 10\)
• Simplify: \(-\frac{5}{10} - \frac{25}{10}i = -\frac{1}{2} - \frac{5}{2}i\)
Answer: \(-\frac{1}{2} - \frac{5}{2}i\)
Complex Numbers Mastery • Teacher Solutions Answer Key • Page 2 of 5
Answer Key Part 5: Equations & Square Root Property
Algebra II
5
#13 \((3x-5)+(2y+7)i = 10-3i\)
• Real: \(3x - 5 = 10 \implies 3x = 15 \implies x = 5\)
• Imag: \(2y + 7 = -3 \implies 2y = -10 \implies y = -5\)
Solution: \(x = 5,\; y = -5\)
#14 \(3x^2 + 48 = 0\)
• Subtract 48: \(3x^2 = -48\)
• Divide by 3: \(x^2 = -16\)
• Square root property: \(x = \pm \sqrt{-16} = \pm 4i\)
Answer: \(x = \pm 4i\)
#15 \((x + 7)^2 + 64 = 0\)
• Subtract 64: \((x + 7)^2 = -64\)
• Square root: \(x + 7 = \pm \sqrt{-64} = \pm 8i\)
• Subtract 7: \(x = -7 \pm 8i\)
Answer: \(x = -7 \pm 8i\)
5
#16 \(4x^2 + 144 = 0\)
• Subtract 144 from both sides:
\(4x^2 = -144\)
• Divide by 4:
\(x^2 = -36\)
• Apply square root property:
\(x = \pm \sqrt{-36} = \pm 6i\)
Answer: \(x = \pm 6i\)
#17 \(2(x - 3)^2 + 32 = 0\)
• Subtract 32: \(2(x - 3)^2 = -32\)
• Divide by 2: \((x - 3)^2 = -16\)
• Extract square roots: \(x - 3 = \pm 4i\)
• Add 3: \(x = 3 \pm 4i\)
Answer: \(x = 3 \pm 4i\)
Complex Numbers Mastery • Teacher Solutions Answer Key • Page 3 of 5
Answer Key Part 6: Completing the Square Solutions (Problems 18-21)
Algebra II
#18 \(x^2 + 6x + 34 = 0\)
1. Isolate: \(x^2 + 6x = -34\)
2. \((b/2)^2 = (6/2)^2 = 3^2 = 9\)
3. Add 9: \(x^2 + 6x + 9 = -34 + 9 = -25\)
4. Factor: \((x + 3)^2 = -25\)
5. Root: \(x + 3 = \pm \sqrt{-25} = \pm 5i\)
Solution: \(x = -3 \pm 5i\)
#19 \(x^2 - 10x + 61 = 0\)
1. Isolate: \(x^2 - 10x = -61\)
2. \((b/2)^2 = (-10/2)^2 = (-5)^2 = 25\)
3. Add 25: \(x^2 - 10x + 25 = -61 + 25 = -36\)
4. Factor: \((x - 5)^2 = -36\)
5. Root: \(x - 5 = \pm \sqrt{-36} = \pm 6i\)
Solution: \(x = 5 \pm 6i\)
#20 \(x^2 + 4x + 29 = 0\)
1. Isolate: \(x^2 + 4x = -29\)
2. \((b/2)^2 = (4/2)^2 = 2^2 = 4\)
3. Add 4: \(x^2 + 4x + 4 = -29 + 4 = -25\)
4. Factor: \((x + 2)^2 = -25\)
5. Root: \(x + 2 = \pm 5i \implies x = -2 \pm 5i\)
Solution: \(x = -2 \pm 5i\)
#21 \(x^2 - 12x + 72 = 0\)
1. Isolate: \(x^2 - 12x = -72\)
2. \((b/2)^2 = (-12/2)^2 = (-6)^2 = 36\)
3. Add 36: \(x^2 - 12x + 36 = -72 + 36 = -36\)
4. Factor: \((x - 6)^2 = -36\)
5. Root: \(x - 6 = \pm 6i \implies x = 6 \pm 6i\)
Solution: \(x = 6 \pm 6i\)
Complex Numbers Mastery • Teacher Solutions Answer Key • Page 4 of 5
Answer Key Part 6 (Cont.): Non-Unit Leading Coefficients (Problems 22-25)
Algebra II
#22 \(x^2 + 8x + 25 = 0\)
1. Isolate: \(x^2 + 8x = -25\)
2. \((b/2)^2 = (8/2)^2 = 4^2 = 16\)
3. Add 16: \(x^2 + 8x + 16 = -25 + 16 = -9\)
4. Factor: \((x + 4)^2 = -9\)
5. Root: \(x + 4 = \pm 3i \implies x = -4 \pm 3i\)
Solution: \(x = -4 \pm 3i\)
#23 \(2x^2 + x + 1 = 0\)
\(a = 2\)
1. Divide by 2: \(x^2 + \frac{1}{2}x = -\frac{1}{2}\)
2. Add \((1/4)^2 = \frac{1}{16}\): \(x^2 + \frac{1}{2}x + \frac{1}{16} = -\frac{8}{16} + \frac{1}{16}\)
3. Combine: \((x + \frac{1}{4})^2 = -\frac{7}{16}\)
4. Extract root: \(x + \frac{1}{4} = \pm \sqrt{-\frac{7}{16}} = \pm \frac{i\sqrt{7}}{4}\)
Exact Solution: \(x = -\frac{1}{4} \pm \frac{\sqrt{7}}{4}i\)
#24 \(3x^2 + 18x + 75 = 0\)
\(a = 3\)
1. Divide by 3: \(x^2 + 6x + 25 = 0 \implies x^2 + 6x = -25\)
2. Add \((6/2)^2 = 9\): \(x^2 + 6x + 9 = -25 + 9 = -16\)
3. Factor: \((x + 3)^2 = -16\)
4. Root: \(x + 3 = \pm 4i \implies x = -3 \pm 4i\)
Exact Solution: \(x = -3 \pm 4i\)
#25 \(5x^2 + 2x + 1 = 0\)
\(a = 5\)
1. Divide by 5: \(x^2 + \frac{2}{5}x = -\frac{1}{5}\)
2. Add \((1/5)^2 = \frac{1}{25}\): \(x^2 + \frac{2}{5}x + \frac{1}{25} = -\frac{5}{25} + \frac{1}{25}\)
3. Combine: \((x + \frac{1}{5})^2 = -\frac{4}{25}\)
4. Extract root: \(x + \frac{1}{5} = \pm \frac{2i}{5} \implies x = -\frac{1}{5} \pm \frac{2}{5}i\)
Exact Solution: \(x = -\frac{1}{5} \pm \frac{2}{5}i\)
Complex Numbers Mastery • Teacher Solutions Answer Key • Page 5 of 5
