Question

All possible roots of x^3+7x^2-12x-18

Answer 1

Solve for x: by completing the square:

x^3 + 7 x^2 - 12 x - 18 = 0

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

(x + 1) (x^2 + 6 x - 18) = 0

Split into two equations:

x + 1 = 0 or x^2 + 6 x - 18 = 0

Subtract 1 from both sides:

x = -1 or x^2 + 6 x - 18 = 0

Add 18 to both sides:

x = -1 or x^2 + 6 x = 18

Add 9 to both sides:

x = -1 or x^2 + 6 x + 9 = 27

Write the left hand side as a square:

x = -1 or (x + 3)^2 = 27

Take the square root of both sides:

x = -1 or x + 3 = 3 sqrt(3) or x + 3 = -3 sqrt(3)

Subtract 3 from both sides:

x = -1 or x = 3 sqrt(3) - 3 or x + 3 = -3 sqrt(3)

Subtract 3 from both sides:

Answer:  x = -1 or x = 3 sqrt(3) - 3 or x = -3 - 3 sqrt(3)