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.