Hello!
First you find the least common multiplier which is 2x(x + 2)
[tex] \frac{1}{x} * 2x(x + 2) + \frac{1}{x + 2} * 2x(x + 2) = \frac{1}{2} * 2x(x + 2)[/tex]
Simplify
2(x + 2) + 2x = x(x + 2)
Now you solve it algebraically
Distribute the 2
2x + 4 + 2x = x(x + 2)
Distribute the x
[tex]2x + 4 + 2x = x^{2} +2x[/tex]
Combine like terms
[tex]4x + 4 = x^{2} +2x[/tex]
Subtract 4 from both sides
[tex] x^{2} + 2x - 4 = 4x[/tex]
Subtract 4x from both sides
[tex] x^{2} - 2x - 4 = 0[/tex]
Now you put this into the quadratic formula
[tex]x = \frac{-(-2) + and - \sqrt{(-2)^{2}- 4 * 1 * (-4) } }{2 * 1} [/tex]
Simplify
[tex]x = \frac{2 + and - \sqrt{20} }{2} [/tex]
Factor the sqrt of 20
[tex]x = \frac{2 + and -2 \sqrt{5} }{2} [/tex]
Factor 2 + and - the 2 * sqrt of 5
[tex] \frac{2(1 + and - \sqrt{5}) }{2} [/tex]
The 2's cancel each other other
[tex]x = 1 + and - \sqrt{5} [/tex]
The answers are [tex]x = 1 + \sqrt{5} and x = 1- \sqrt{5} [/tex]
Hope this helps!
