Question

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

Answer 1

The answers are `-3, 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+5/y=12/(y^2)`
`2y^2+5y=12`
`2y^2+5y-12=0`
`y=(-5+-sqrt(5^2-4(2)(-12)))/(2(2))`
`=(-5+-sqrt(25+96))/4`
`=(-5+-sqrt(121))/4`
`=(-5+-11)/4`
`y=-4,3/2`

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

`r=-3,5/2`

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

If you expand, you would have got:

`2r^2+r-15=0`

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