Question

How do you solve `2^ { 3} = \frac { x ^ { 2} } { 6- x }`?

Answer 1

See a solution process below:

Explanation:

First, evaluate the term on the left side of the equation:

`2^3 = x^2/(6 - x)`

`8 = x^2/(6 - x)`

Next, multiply each side of the equation by `color(red)((6 - x))` to eliminate the fraction while keeping the equation balanced:

`8color(red)((6 - x)) = x^2/(6 - x) xx color(red)((6 - x))`

`(8 xx color(red)(6)) - (8 xx color(red)(x)) = x^2/color(red)(cancel(color(black)(6 - x))) xx cancel(color(red)((6 - x)))`

`48 - 8x = x^2`

Now, subtract `color(red)(48)` and add `color(blue)(8x)` to each side of the equation to put the equation in standard form while keeping the equation balanced:

`48 - color(red)(48) - 8x + color(blue)(8x) = x^2 + color(blue)(8x) - color(red)(48)`

`0 -0 = x^2 + 8x - 48`

`0 = x^2 + 8x - 48`

`x^2 + 8x - 48 = 0`

Next, factor the left side of the equation as:

#(x + 12)(x - 4) = 0

Now, solve each term on the left for `0`

Solution 1:

`x + 12 = 0`

`x + 12 - color(red)(12) = 0 - color(red)(12)`

`x + 0 = -12`

`x = -12`

Solution 2:

`x - 4 = 0`

`x - 4 + color(red)(4) = 0 + color(red)(4)`

`x - 0 = 4`

`x = 4`

The Solution Is: `x = {-12, 4}`