13. f (x) = x^4

- 9x2 - 22x - 24

x = 4 or x = - 2

Step-by-step explanation:

Solve for x over the real numbers:

x^4 - 9 x^2 - 22 x - 24 = 0

The left hand side factors into a product with three terms:

(x - 4) (x + 2) (x^2 + 2 x + 3) = 0

Split into three equations:

x - 4 = 0 or x + 2 = 0 or x^2 + 2 x + 3 = 0

Add 4 to both sides:

x = 4 or x + 2 = 0 or x^2 + 2 x + 3 = 0

Subtract 2 from both sides:

x = 4 or x = - 2 or x^2 + 2 x + 3 = 0

Subtract 3 from both sides:

x = 4 or x = - 2 or x^2 + 2 x = - 3

Add 1 to both sides:

x = 4 or x = - 2 or x^2 + 2 x + 1 = - 2

Write the left hand side as a square:

x = 4 or x = - 2 or (x + 1) ^2 = - 2

(x + 1) ^2 = - 2 has no solution since for all x on the real line, (x + 1) ^2 >=0 and - 2<0:

Answer: x = 4 or x = - 2