Hey there!
First, you must find the common denominator of the equation.
To find the common denominator, first take a look at all the factors of each denominator:
x: x
x+4: x+4
6: 2*3
Next, because you do not have a common multiple in the denominators, you would multiply the denominators together to create one common multiple:
6(x)(x+4). This would be used to remove the fractions from the equation to make it easier to solve.
Now, multiply the common multiple 6(x)(x+4) to the entire equation:
(6(x)(x+4)) × (
![\frac{1}{x}](https://tex.z-dn.net/?f=%20%5Cfrac%7B1%7D%7Bx%7D%20)
+
![\frac{1}{x+4}](https://tex.z-dn.net/?f=%20%5Cfrac%7B1%7D%7Bx%2B4%7D%20)
=
![\frac{1}{6}](https://tex.z-dn.net/?f=%20%5Cfrac%7B1%7D%7B6%7D%20)
)
When you multiply the factor to the equation, the x in the common factor would cancel out the x in
![\frac{1}{x}](https://tex.z-dn.net/?f=%20%5Cfrac%7B1%7D%7Bx%7D%20)
resulting in just 6(x+4).
The x+4 would cancel out the x+4 factor in
![\frac{1}{x+4}](https://tex.z-dn.net/?f=%20%5Cfrac%7B1%7D%7Bx%2B4%7D%20)
resulting in just 6x.
The 6 would cancel out the 6 in
![\frac{1}{6}](https://tex.z-dn.net/?f=%20%5Cfrac%7B1%7D%7B6%7D%20)
resulting in just x(x+4).
As a result, when you multiply the common factor 6(x)(x+4) to
![\frac{1}{x}](https://tex.z-dn.net/?f=%20%5Cfrac%7B1%7D%7Bx%7D%20)
+
![\frac{1}{x+4}](https://tex.z-dn.net/?f=%20%5Cfrac%7B1%7D%7Bx%2B4%7D%20)
=
![\frac{1}{6}](https://tex.z-dn.net/?f=%20%5Cfrac%7B1%7D%7B6%7D%20)
, you will get
(6(x+4))+ 6x = x(x+4)
Now, simplify the equation further:
(6(x+4))+ 6x = x(x+4)
6x+24+6x=x^2+4x (I have distributed the values in the parentheses)
12x+24=x^2 + 4x (I combined like terms on both sides)
0=x^2-8x-24 (I have moved all terms to one side so that the x values can be solved for using the quadratic formula)
Because this quadratic does not factor evenly, we must use the quadratic formula in order to find the exact x values:
x=
![\frac{-b±\sqrt{b^2-4ac} }{2a}](https://tex.z-dn.net/?f=%20%5Cfrac%7B-b%C2%B1%5Csqrt%7Bb%5E2-4ac%7D%20%7D%7B2a%7D%20)
Your a value is 1, your b value is -8, and your c value is -24:
x=
![\frac{-(-8) ±\sqrt{(-8)^2-4(1)(-24)} }{2(1)}](https://tex.z-dn.net/?f=%20%5Cfrac%7B-%28-8%29%20%C2%B1%5Csqrt%7B%28-8%29%5E2-4%281%29%28-24%29%7D%20%7D%7B2%281%29%7D%20)
x=
![\frac{8 ±\sqrt{64+96} }{2}](https://tex.z-dn.net/?f=%20%5Cfrac%7B8%20%C2%B1%5Csqrt%7B64%2B96%7D%20%7D%7B2%7D%20)
x=
![\frac{8 ±\sqrt{160} }{2}](https://tex.z-dn.net/?f=%20%5Cfrac%7B8%20%C2%B1%5Csqrt%7B160%7D%20%7D%7B2%7D%20)
x=
![\frac{8 ±4\sqrt{10} }{2}](https://tex.z-dn.net/?f=%20%5Cfrac%7B8%20%C2%B14%5Csqrt%7B10%7D%20%7D%7B2%7D%20)
x=4±2
![\sqrt{10}](https://tex.z-dn.net/?f=%20%5Csqrt%7B10%7D%20)
Therefore, your x values are 4+2
![\sqrt{10}](https://tex.z-dn.net/?f=%20%5Csqrt%7B10%7D%20)
and 4-2