Question

6. What are the zeroes for thefunctionf(x) = 2x3 + 12x – 10x2?​

Answer 1

Answer:

x = 0, x = 5 + √13 and x = 5 - √13.

Step-by-step explanation:

f(x) = 2x^3 + 12x – 10x^2 can and should be rewritten in descending powers of x:  

f(x) = 2x^3 – 10x^2 + 12x

This, in turn, can be factored into f(x) = x·(x² - 10x + 12).

Setting this last result = to 0 results in f(x) = x·(x² - 10x + 12).

Thus, x = 0 is one root.  Two more roots come from x² - 10x + 12 = 0.

Let's "complete the square" to solve this equation.

Rewrite x² - 10x + 12 = 0  as  x² - 10x +              12 = 0.

a) Identify the coefficient of the x term.  It is -10.

b) take half of this result:  -5

c) square this last result:  (-5)² = 25.

d) Add this 25 to both sides of x² - 10x +              12 = 0:

     x² - 10x +  25      +      12 = 0 + 25

e) rewrite x² - 10x +  25 as the square of a binomial:

       (x - 5)² = 13

f)  taking the sqrt of both sides:  x - 5 = ±√13

g) write out the zeros:  x = 5 + √13 and x = 5 - √13.

The three roots are x = 0, x = 5 + √13 and x = 5 - √13.