F(x) = 10x3 – 13x² + 12x + 3

x = -1/5

Step-by-step explanation:

Solve for x over the real numbers:

10 x^3 - 13 x^2 + 12 x + 3 = 0

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

(5 x + 1) (2 x^2 - 3 x + 3) = 0

Split into two equations:

5 x + 1 = 0 or 2 x^2 - 3 x + 3 = 0

Subtract 1 from both sides:

5 x = -1 or 2 x^2 - 3 x + 3 = 0

Divide both sides by 5:

x = -1/5 or 2 x^2 - 3 x + 3 = 0

Divide both sides by 2:

x = -1/5 or x^2 - (3 x)/2 + 3/2 = 0

Subtract 3/2 from both sides:

x = -1/5 or x^2 - (3 x)/2 = -3/2

Add 9/16 to both sides:

x = -1/5 or x^2 - (3 x)/2 + 9/16 = -15/16

Write the left hand side as a square:

x = -1/5 or (x - 3/4)^2 = -15/16

(x - 3/4)^2 = -15/16 has no solution since for all x on the real line, (x - 3/4)^2 >=0 and -15/16<0:

x = -1/5