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

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

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

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 \[(5-4)-5(5-4)^\frac{1}{2}=1-5(1)^\frac{1}{2}=1-5(1)=1-5=-4 \neq 6\]

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

Join our real-time social learning platform and learn together with your friends!