Question

Find all x-intercepts of the function. Express your answer as a list of x-values. f ( x ) = − 45 x 3 + 60 x 2 − 20 x

Answer 1

Answer:

x = 0, $\frac{2}{3}$

Step-by-step explanation:

Factorise f(x) and equate to zero for x- intercepts

f(x) = - 5x(9x² - 12x + 4) = 0 ( common factor of - 5x )

To factor the quadratic consider the factors of the product of the x² term and the constant term which sum to give the coefficient of the x- term, that is

product = 9 × 4 = 36 , sum = - 12

The factors are - 6 and - 6

Use these factors to split the middle term

9x² - 6x - 6x + 4 = 0 ( factor the first/second and third/fourth terms )

3x(3x - 2) - 2(3x - 2)  ( factor out (3x - 2) )

(3x - 2)(3x - 2), hence

f(x) = - 5x(3x - 2)(3x - 2) = 0

equate each factor to zero and solve for x

- 5x = 0 ⇒ x = 0

(3x - 2)² = 0 ⇒ x = $\frac{2}{3}$ of multiplicity 2, indicating turning point

x - intercepts are x = 0, $\frac{2}{3}$