Question

2/x-5=4x

What is the value of x

Answer 1

`\large\boxed{x=(5\pm3\sqrt3)/(2)}`

Explanation:

`Domain:\ x-5\\eq0\to x\\eq5\\\\(2)/(x-5)=4x\\\\(2)/(x-5)=(4x)/(1)\qquad\text{cross multiply}\\\\(4x)(x-5)=(2)(1)\qquad\text{use the distributive property}\\\\(4x)(x)+(4x)(-5)=2\\\\4x^2-20x=2\\\\2^2x^2-20x=2\\\\(2x)^2-2(2x)(5)=2\qquad\text{add}\ 5^2\ \text{to both sides}\\\\(2x)^2-2(2x)(5)+5^2=2+5^2\qquad\text{use}\ (a-b)^2=a^2-2ab+b^2\\\\(2x-5)^2=2+25\\\\(2x-5)^2=27\to2x-5=\pm√(27)\qquad\text{add 5 to both sides}`

`2x=5\pm√(9\cdot3)\qquad\text{use}\ √(ab)=√(a)\cdot√(b)\\\\2x=5\pm\sqrt9\cdot\sqrt3\\\\2x=5\pm3\sqrt3\qquad\text{divide both sides by 2}\\\\x=(5\pm3\sqrt3)/(2)`