Question

How do I solve this equation?


`2x( {4 - x)}^{ - (1 )/(2) } - 3 {(4 - x)}^{ (1)/(2) } = 0`

Answer 1

`2x(4-x)^(-1/2)-3(4-x)^(1/2)=(4-x)^(-1/2)\bigg(2x-3(4-x)\bigg)`

We effectively rewrite the equation as

`(5x-12)/((4-x)^(1/2))=0`

In order for the LHS to be defined, we need to restrict
`4-x>0`, or
`x<4`. Now, the LHS will vanish when the numerator is 0, which happens for

`5x-12=0\implies5x=12\implies x=\frac{12}5`

This value is indeed smaller than 4, so the solution is
`x=\frac{12}5`.