Find the greatest value of t such that \(\frac{t^2 - t -56}{t-8} = \frac{3}{t+5}.\)
Solve for t:
(t^2 - t - 56)/(t - 8) = 3/(t + 5)
Cross multiply:
(t + 5) (t^2 - t - 56) = 3 (t - 8)
Expand out terms of the left hand side:
t^3 + 4 t^2 - 61 t - 280 = 3 (t - 8)
Expand out terms of the right hand side:
t^3 + 4 t^2 - 61 t - 280 = 3 t - 24
Subtract 3 t - 24 from both sides:
t^3 + 4 t^2 - 64 t - 256 = 0
The left hand side factors into a product with three terms:
(t - 8) (t + 4) (t + 8) = 0
Split into three equations:
t - 8 = 0 or t + 4 = 0 or t + 8 = 0
Add 8 to both sides:
t = 8 or t + 4 = 0 or t + 8 = 0
Subtract 4 from both sides:
t = 8 or t = -4 or t + 8 = 0
Subtract 8 from both sides:
t = 8 or t = -4 or t = -8
(t^2 - t - 56)/(t - 8) ⇒ (-56 - -8 + (-8)^2)/(-8 - 8) = -1
3/(t + 5) ⇒ 3/(5 - 8) = -1:
So this solution is correct
(t^2 - t - 56)/(t - 8) ⇒ (-56 - -4 + (-4)^2)/(-8 - 4) = 3
3/(t + 5) ⇒ 3/(5 - 4) = 3:
So this solution is correct
(t^2 - t - 56)/(t - 8) ⇒ (-56 - 8 + 8^2)/(8 - 8) = (undefined)
3/(t + 5) ⇒ 3/(5 + 8) = 3/13:
So this solution is incorrect
The solutions are:
t = -4 or t = -8
+36916 Factor the right numerator to get:
(t-8)(t+7) / (t-8) = 3/(t+5) the term (t-8) cancles out on the right but REMEMBER t cannot equal 8 (would make zero denominator)
t+7 = 3/ (t+5)
(t+7)*(t+5) = 3
t^2 + 12t + 35 = 3
t^2 + 12t + 32 = 0
(t+8)(t+4) = 0 Shows t = -8 or -4 the greatest of which is -4
