What is x squared minus x minus 6 divided by x squared minus 4?

Answer 1

2

Answer 2

f(x) = (x2 - x - 6)/(x2 - 4)

To solve this function, we must first obtain the simplified binomial form. This can be conceived as the reverse operation of FOILing:

(x2 - x - 6) = (x + _) (x - _)

Since we see the negative value on the constant 6, we know we must have a positive and negative for each of the respective binomials as a negative and a positive multiplied is always negative. From the scalar of the x term, we see that it is negative and knowing how FOIL works (multiply First, Outer, Inner, Last, then add like terms) we know that the negative value must be larger. Since we know the constant is 6, we must choose terms which multiply to give us six while remembering the negative term must be the larger of the two:

(x2 - x - 6) = (x + 2) (x - 3).

We must do the same for the bottom of the equation:

(x2 - 4) = (x + _) (x - _)

(x2 - 4) = (x + 2) (x - 2)

Now we have simplifed the equation and may solve it:

f(x) = (x + 2) (x - 3) / (x + 2) (x - 2)

Like terms will be reduced, as dividing 1/1 = 1:

f(x) = (x - 3) / (x - 2)

We then must determine where the function f(x) = 0.

(x - 3) = 0 @ x = 3

(x - 2) = 0 @ x = 2 Undefined

The second term fails however, as it would result in an undefined function since division by 0 is impossible.

This provides us with the x-intercepts of the function. You should then check your answer by replacing x in the orignial equation with the x-intercept.

f(x) = (x2 - x - 6)/(x2 - 4)

f(x) = (32 - 3 - 6)/(32 - 4)

f(x) = (9 - 3 - 6)/(9 - 4)

f(x) = (0)/(5)

f(x) = 0

Always check your work and show each step.