Answer:
x = 0, x = 5 + √13 and x = 5 - √13.
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.