Mathematics
OpenStudy (anonymous):

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

OpenStudy (saifoo.khan):

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

myininaya (myininaya):

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

myininaya (myininaya):

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

myininaya (myininaya):

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

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