Question

How do you solve `\frac { x ^ { 2} - 16} { x + 4} = x - 4`?

Answer 1

See a solution process below:

Explanation:

The numerator of the fraction on the left side of the equation is a special form of a quadratic.

`(a - b)(a + b) = a^2 - b^2`

Letting `a^2 = x` and `b^2 = 16` we can factor the numerator as:

`((x + 4)(x - 4))/(x + 4) = (x - 4)`

We can now cancel like terms in the numerator and denominator:

`(color(red)(cancel(color(black)((x + 4))))(x - 4))/color(red)(cancel(color(black)(x + 4))) = (x - 4)`

`x - 4 = x - 4`

We can now add `color(red)(4)` to each side of the equation:

`x - 4 + color(red)(4) = x - 4 + color(red)(4)`

`x - 0 = x - 0`

`x = x`

This means `x` can be any value and this equation will be true.

Except! from the original equation we cannot divide by `0`, therefore:

`x + 4 != 0` or `x != -4`