Find all values of $t$ that satisfy $\dfrac{t+4}{t+5} = \dfrac{t-5}{2t}$.
Solve for t:
(t + 4)/(t + 5) = (t - 5)/(2 t)
Cross multiply:
2 t (t + 4) = (t - 5) (t + 5)
Expand out terms of the left hand side:
2 t^2 + 8 t = (t - 5) (t + 5)
Expand out terms of the right hand side:
2 t^2 + 8 t = t^2 - 25
Subtract t^2 - 25 from both sides:
t^2 + 8 t + 25 = 0
Subtract 25 from both sides:
t^2 + 8 t = -25
Add 16 to both sides:
t^2 + 8 t + 16 = -9
Write the left hand side as a square:
(t + 4)^2 = -9
Take the square root of both sides:
t + 4 = 3 i or t + 4 = -3 i
Subtract 4 from both sides:
t = -4 + 3 i or t + 4 = -3 i
Subtract 4 from both sides:
t = -4 + 3 i or t = -4 - 3 i
