**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.