Help please stumped math?

Question

How to find break-even points of profit function?

Which of the following profit functions has break-even points at x = 3 and x = 11?

a) P(x) = -2 (x^2 - 28x + 66)

b) P(x) = -2 (x + 3)(x + 11)

c) P(x) = -2 (x - 3)(x + 11)

d) P(x) = -2(x^2 - 14x + 33)

I have tried to solve this by completing the square for a and d because I don't think b and c are possible answers and it did not work out I tried and tried again. If someone knows the answer could you please show/explain the steps in addition to naming the appropriate letter answer. Thanks in advance.

Answer 1

d)

P(x) = -2 * (x^2 - 14x + 33)

To factor the trinomial into (x + a)(x + b), we need to identify two numbers (a & b) whose sum is -14 and whose product is +33.

a + b = -14 so b = -14 - a

ab = +33

substitute (-14 - a) for b in ab = +33

a (-14 - a) = 33

-14a - a² = 33

multiply both sides by -1

14a + a² = -33

a² + 14a = -33

complete the square by taking 1/2 of 14 and squaring it (=49)

a² + 14a + 49 = -33 + 49

(a + 7)² = 16

take square root of both sides

a + 7 = 4

a = -3

and since ab = 33, then b = -11

so, our factors are

P(x) = -2 * (x - 11) * (x - 3)

Therefore, x = 11 and x = 3.