# How do I use the quadratic formula to solve 2 + 5/(r-1) = 12/((r-1)^2)?

Aug 22, 2014

The answers are $- 3 , \frac{5}{2}$.

You may be tempted to expand $r - 1$, but let's use substitution: $y = r - 1$

$r = y + 1$

We have an NPV: $r = 1$

$2 + \frac{5}{y} = \frac{12}{{y}^{2}}$
$2 {y}^{2} + 5 y = 12$
$2 {y}^{2} + 5 y - 12 = 0$
$y = \frac{- 5 \pm \sqrt{{5}^{2} - 4 \left(2\right) \left(- 12\right)}}{2 \left(2\right)}$
$= \frac{- 5 \pm \sqrt{25 + 96}}{4}$
$= \frac{- 5 \pm \sqrt{121}}{4}$
$= \frac{- 5 \pm 11}{4}$
$y = - 4 , \frac{3}{2}$

You may be excited to get an answer and stop, but remember that we used substitution, so substitute back: $r = y + 1$

$r = - 3 , \frac{5}{2}$

This does not conflict with the NPV, so the answer is good.

If you expand, you would have got:

$2 {r}^{2} + r - 15 = 0$

Then you would use the quadratic and the answer would be the same.