Answer:
x = 4/5
Step-by-step explanation:
We are given the equation:
[tex] \displaystyle \large{ \frac{5x - 4}{ \sqrt{5x} + 2 } = 2 - \frac{ \sqrt{5x} + 2}{2} }[/tex]
Multiply both sides by LCM which is 2(√5x +2) to clear out the denominator.
[tex] \displaystyle \large{ \frac{5x - 4}{ \sqrt{5x} + 2 }(2)(\sqrt{5x} + 2)= 2 (2)( \sqrt{5x} + 2) - \frac{ \sqrt{5x} + 2}{2} (2)( \sqrt{5x + 2} )} \\ \displaystyle \large{ (5x - 4)2= 4( \sqrt{5x} + 2) - (\sqrt{5x} + 2)( \sqrt{5x} + 2)} \\ \displaystyle \large{ 10x - 8= 4\sqrt{5x} + 8 - ( \sqrt{5x} + 2) ^{2} } \\ \displaystyle \large{ 10x - 8= 4\sqrt{5x} + 8 - (5x + 4 \sqrt{5x} + 4) } \\ \displaystyle \large{ 10x - 8= 4\sqrt{5x} + 8 - 5x - 4 \sqrt{5x} - 4} \\ \displaystyle \large{ 10x - 8= 4 - 5x}[/tex]
Thus, our simplified equation is;-
[tex] \displaystyle \large{10x - 8 = 4 - 5x}[/tex]
Add both sides by 5x then add both sides by 8.
[tex] \displaystyle \large{10x + 5x - 8 = 4 - 5x + 5x} \\ \displaystyle \large{15x - 8 = 4} \\ \displaystyle \large{15x - 8 + 8 = 4 + 8} \\ \displaystyle \large{15x= 12}[/tex]
Divide both sides by 15.
[tex] \displaystyle \large{ \frac{15x}{15} = \frac{12}{15} } \\ \displaystyle \large{x = \frac{4}{5} }[/tex]
Therefore, x = 4/5
