Help in rearranging terms

Question

Given that Q=W/(1-(T₀/T)) and Q=a(T-T₀)

rearrange so that T is the subject.

The final answer is:

T=T₀ + (W/2a) + (1 + sqrt(1+(4aT₀/W))

But I just don't know how to get to it...

Answer 1

Given that Q=W/(1-(T0/T)) and Q=a(T-T0)

W/(1-(T0/T)) = a(T-T0)

W=(1-(T0/T))• a(T-T0)

WT = a(T-T0)^2

0=aT^2-(a2T0+W)T+aT0^2

 

Using quadratic formula

T=T0+W/(2a)+-sqrt((a2T0+W)^2-(2aT0)^2)/2a

T=T0+W/(2a)+-sqrt((T0+W/2a)^2-(T0)^2)

T=T0+W/(2a)+-sqrt((T0W/a+(W/2a)^2)

T=T0+W/(2a)+-W/2a•sqrt((4aT0/W+1)

T=T0+W/(2a)(1+-sqrt((4aT0/W+1)

 

I suspect the real world causes the discard if the - solution. But I don't know?

 

This differs from the answer given by a plus sign. 

Answer 2

This one is nasty. OK, that's a safe statement, but not nearly as nasty as I made it. Hats off to Amy M.

 

I didn't see where

 

WT = a(1 - T₀/T)(T - T₀) = a[(T - T₀)/T](T - T₀), which allows you to multiply through by T and move on to get

 

WT = a(T - T₀)².

 

Nice.