State the Domain and show work.

y = x + 2 / x^2 - 4x - 12

To determine the domain of the function y = (x + 2) / (x^2 - 4x - 12), we need to find all the values of x for which the function is defined.

First, let's determine the values of x that would make the function undefined. This occurs when the denominator is equal to zero because division by zero is undefined in mathematics.

Setting the denominator equal to zero:
x^2 - 4x - 12 = 0

We can solve this quadratic equation by factoring or using the quadratic formula. Factoring may not always be straightforward, so let's use the quadratic formula here:

x = (-b ± √(b^2 - 4ac)) / 2a

For the equation x^2 - 4x - 12 = 0, we have:
a = 1, b = -4, c = -12

x = (-(-4) ± √((-4)^2 - 4(1)(-12))) / 2(1)
x = (4 ± √(16 + 48)) / 2
x = (4 ± √64) / 2
x = (4 ± 8) / 2

This gives us two possible solutions for x:
x1 = (4 + 8) / 2 = 12 / 2 = 6
x2 = (4 - 8) / 2 = -4 / 2 = -2

Now, we have found the x-values for which the denominator equals zero: x = 6 and x = -2.

To find the domain, we need to exclude these values since the function is undefined at those points. Therefore, the domain of the function y = (x + 2) / (x^2 - 4x - 12) is all real numbers except x = 6 and x = -2.

The domain is what x values you can successfully input.