Question

solve (x-4)-5(x-4)^(1/2)=6

Answer 1

\[(x-4) -5\sqrt{x-4}=6\]

Answer 2

first it may be easier to replace x-4 with u for now so we have
\[u-5u^\frac{1}{2}=6 \]
ok but if we let \[s=u^\frac{1}{2}=>s^2=u\]
so we have
\[s^2-5s=6 =>s^2-5s-6=0 => (s-6)(s+1)=0 =>s=6 , s=-1\]
but \[s=u^\frac{1}{2}\]
so we have
\[u^\frac{1}{2}=6, u^\frac{1}{2}=-1 =>u=36, u=1 \]
but remember we replace x-4 with u
so we actually have
\[x-4=36, x-4=1 => x=36+4=40, x=1+4=5\]
now we must check these
whenever you raised both sides to even power could give you extra solutions

Answer 3

so we have x=40;
\[(40-4)-5(40-4)^\frac{1}{2}=36-5(36)^\frac{1}{2}=36-5(6)=36-30=6\]
so x=40 works;
so we have x=5
\[(5-4)-5(5-4)^\frac{1}{2}=1-5(1)^\frac{1}{2}=1-5(1)=1-5=-4 \neq 6\]

Answer 4

so x=5 does not work
and the only solution is x=40