How do you evaluate \frac{1}{x^{2}-16}=-\frac{1}{x^{2}-4x}?

2 Answers
Dec 16, 2017

See a solution process below:

Explanation:

First, factor the denominators of each fraction as:

1/((x + 4)(x - 4)) = -1/(x (x - 4)

Next, multiply each side of the equation by color(red)((x - 4)) to eliminate a common factor while keeping the equation balanced:

color(red)((x - 4)) xx 1/((x + 4)(x - 4)) = color(red)((x - 4)) xx -1/(x(x - 4)

cancel(color(red)((x - 4))) xx 1/((x + 4)color(red)(cancel(color(black)((x - 4))))) = cancel(color(red)((x - 4))) xx -1/(xcolor(red)(cancel(color(black)((x - 4)))))

1/(x + 4) = -1/x

Then, because we have a pure fraction on each side of the equation we can flip the fractions giving:

(x + 4)/1 = -x/1

x + 4 = -x

Next, we can subtract color(red)(4) and add color(blue)(x) to each side of the equation to isolate the x term while keeping the equation balanced:

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

1color(blue)(x) + 1x + 0 = 0 - 4

(1 + 1)x = -4

2x = -4

Now, divide both sides of the equation by color(red)(2) to solve for x while keeping the equation balanced:

(2x)/color(red)(2) = -4/color(red)(2)

(color(red)(cancel(color(black)(2)))x)/cancel(color(red)(2)) = -2

x = -2

Dec 16, 2017

x=-2

Explanation:

1/(x^2-16)=-1/(x^2-4x)

We can multiply both sides of the equation by (x^2-16)(x^2-4x)

color(red)(cancel((x^2-16))(x^2-4x))/cancel(x^2-16)=-color(red)((x^2-16)cancel((x^2-4x)))/cancel(x^2-4x)

x^2-4x=-(x^2-16)
x^2-4x=-x^2+16

Solve like a quadratic:

2x^2-4x-16=0
x^2-2x-8=0
(x-4)(x+2)=0

x=-2 or x=4

We need to check our answers however.
This is because we can't divide by zero, this is undefined

Because of this, if we let x=4, we get denominators of 0. This cannot happen, so x!=4 This means that the only solution is x=-2