51:
Solve for x:
x^4-5 x^3+15 x^2-45 x+54 = 0
The left hand side factors into a product with three terms:
(x-3) (x-2) (x^2+9) = 0
Split into three equations:
x-3 = 0 or x-2 = 0 or x^2+9 = 0
Add 3 to both sides:
x = 3 or x-2 = 0 or x^2+9 = 0
Add 2 to both sides:
x = 3 or x = 2 or x^2+9 = 0
Subtract 9 from both sides:
x = 3 or x = 2 or x^2 = -9
Take the square root of both sides:
Answer: |x = 3 or x = 2 or x = 3i or x = -3i
52:
Solve for x:
2 x^4+5 x^3-10 x^2+10 x+8 = 0
The left hand side factors into a product with three terms:
(x+4) (2 x+1) (x^2-2 x+2) = 0
Split into three equations:
x+4 = 0 or 2 x+1 = 0 or x^2-2 x+2 = 0
Subtract 4 from both sides:
x = -4 or 2 x+1 = 0 or x^2-2 x+2 = 0
Subtract 1 from both sides:
x = -4 or 2 x = -1 or x^2-2 x+2 = 0
Divide both sides by 2:
x = -4 or x = -1/2 or x^2-2 x+2 = 0
Subtract 2 from both sides:
x = -4 or x = -1/2 or x^2-2 x = -2
Add 1 to both sides:
x = -4 or x = -1/2 or x^2-2 x+1 = -1
Write the left hand side as a square:
x = -4 or x = -1/2 or (x-1)^2 = -1
Take the square root of both sides:
x = -4 or x = -1/2 or x-1 = i or x-1 = -i
Add 1 to both sides:
x = -4 or x = -1/2 or x = 1+i or x-1 = -i
Add 1 to both sides:
Answer: |x = -4 or x = -1/2 or x = 1+i or x = 1-i
