I need help understanding this SAT math equation. x^2-4 / x^2+5x+6 = 0?

Question

X^2-4 / x^2+5x+6 = 0

ANSWER CHOICES
A. -3
B. -2
C. 2

The answer is 2.

x^2-4
(x-2)(x+2)
Then it says , "Solving this for x gives either 2 or −2. That means either of these values will make our numerator equal zero."
... I don't know why those values makes the numerator equal to zero.

It also says,"Any values that make our denominator zero must be rejected."
(Which has to do with the next step)

NEXT STEP

x^2+5x+6=0
(x+3)(x+2)
x=-3 ; x=-2

"This means that if x is −3 or −2, we end up dividing by zero. That means that −2 cannot be a valid solution, leaving 2 as the only valid answer."

My question is, how do you know if you're dividing by zero or the values given will make the numerator equal zero?

Answer 1

My question is, how do you know if you're dividing by zero or the values given will make the numerator equal zero?



Answer: you test it.



We know the denominator is:



x² + 5x + 6



So set it equal to zero and solve for x. This will tell you which values of x makes the denominator zero, and then therefore cannot be in the solution set:



x² + 5x + 6 = 0

(x + 3)(x + 2) = 0



x = -3 and -2



So neither of these can be a solution.



So now to solve the main equation:



(x² - 4) / (x² + 5x + 6) = 0



Multiply both sides by the denominator gets rid of it on the left side and the right side is still zero:



x² - 4 = 0



Now we can find two values for x:



x² = 4

x = ±2



But since we already said that x cannot be -2 we can throw it out and that leaves only one solution:



x = 2

Answer 2

x² - 4 = 0

x = ± 2



Check

if x = 2 , x² - 4 = 0

if x = - 2 , x² - 4 = 0

Answer 3

The only way a fraction can have a zero value is if the numerator is zero.

That's why they conclude that x must be either 2 or -2.

If you can't see that x=2 and/or x = -2 will make x^2-4 into a zero,

you must repeat elementary algebra, and you should certainly not be taking SAT II in math.



If you can't see that x = -2 and/or x = -3 will make x^2+5x+6 into a zero, again I say, you are way out of your depth, and nowhere near being ready to apply to college.

Answer 4

The fraction is 0 when the numerator is 0 and the denominator is not 0.



x^2 - 4 is a difference in squares, and can be factored as (x - 2)(x + 2).

If x^2 - 4 = 0, then (x - 2)(x + 2) = 0. If a product of two numbers is zero, then at least one of those numbers is zero.

So x - 2 = 0 or x + 2 = 0. Which means x = 2 or -2.



Now we look at the denominator. x^2+5x+6 can be factored using AC method, quadratic formula, or completing the square. The factors are x+3 and x+2. Using the same logic as before, that means x=-3 or x=-2.



x=-2 must be rejected, since that would make the denominator zero. So that leaves x=2 as the only zero.