Q:  Segment NY has a midpoint, R.  NR = x² – x – 2 and RY = 5x – 7.  Find x and the length of NY.

A:  Since R is the midpoint, we know that NR = RY:

x² – x – 2 = 5x – 7

I see an x², so that means it must be a quadratic, right?  Always start by setting equal to zero:

x² – x – 2 = 5x – 7

x² – x – 2 – 5x + 7 = 0

x² – 6x + 5 = 0

Now, time to factor.  Need help with factoring?  First view Factoring 101:

(x      )(x       ) = 0

We have to multiply to get a +5 and combine to get a -6.  Only a + & + or – & – will work to multiply to get a +.  However, it must be a – & – in order to combine to get a -6.  Right?

(x –   )(x –   ) = 0

Now, what two numbers multiply to give +5 and combine to give -6?  -1 and -5!

(x – 1 )(x – 5 ) = 0

So,

(x – 1) = 0 or (x – 5) = 0

x = 1 or 5

2 answers?!?  How can there be 2 answeres?!?  Let’s check… We are measuring lenghts.. So, if x = 1:

RY = 5x – 7 = 5(1) – 7 = 5 – 7 = -2

RY is a length and cannot equal -2 (no such thing as negative length!).  Therefore, x = 5 is our answer!

RY = 5x – 7 = 5(5) – 7 = 25 – 7 = 18.

RY = NY, so…

NY = 18.