1. x/(x - 5) = 4/(x - 4)
x (x - 4) / ((x - 5) (x - 4)) = 4 (x - 5) / ((x - 5) (x - 4))
x (x - 4) = 4 (x - 5)
x² - 4x = 4x - 20
x² - 8x = -20
Solving for x by completing the square gives
x² - 8x + 16 = -4
(x - 4)² = -4
x - 4 = ± 2i
x = 4 + 2i or x = 4 - 2i
2. Since -7 + 2i and -7 - 2i are roots of the given quadratic, we have
z² + bz + c = (z - (-7 + 2i)) (z - (-7 - 2i))
z² + bz + c = (z + (7 - 2i)) (z + (7 + 2i))
z² + bz + c = z² + ((7 - 2i) + (7 + 2i)) z + (7 - 2i) (7 + 2i)
z² + bz + c = z² + 14z + 53
so that b = 14 and c = 53, which makes b + c = 67.
3. If x + y = 4, then
x³ + y³ = 100
x³ + (4 - x)³ = 100
x³ + (64 - 48x + 12x² - x³) = 100
12x² - 48x + 64 = 100
12x² - 48x = 36
x² - 4x = 3
and by completing the square,
x² - 4x + 4 = 7
(x - 2)² = 7
x - 2 = ± 7
x = 2 + 7 or x = 2 - 7
x = 9 or x = -5
If x = 9, then y = -5, so one pair of solutions would be (x, y) = (9, -5).
