I need to simplify the ratio: x^2-x-20 / x^2+8x+16.
I know that the denominator is simplified into (x+4)(x+4), But I'm unsure as to how to work the numerator. Help?

Question

I need to simplify the ratio: x^2-x-20 / x^2+8x+16.   
I know that the denominator is simplified into (x+4)(x+4), But I'm unsure as to how to work the numerator. Help?

anonymous · Answer

Solve for the roots of the first equation, by using the quadratic formula$$x=\frac{-b \pm \sqrt{b^2-4ac}}{2a}=\frac{1 \pm \sqrt{1-4*1*(-20)}}{2}=\frac{1 \pm \sqrt{1+80}}{2}$$You then get,$$x=\frac{1 \pm \sqrt{81}}{2}=\frac{1 \pm 9}{2}$$From there you have two solutions x1=-4, and x2=5.

Now the equation$$x^2-x-20 = (x-x1)(x-x2)$$So just replace the x1 and x2.$$(x-5)(x+4)$$

anonymous · Answer

Alright, but isn't that solving the equation? I just need to simplify. I'm thinking I can write it out into x+x+x+4*5, and then use the x+4 in the numerator to cancel out one from the denominator, which would leave us with 5x^2 / x+4, but I'm not sure if that's correct.

anonymous · Answer

No no, you get this$$\frac{(x-5)(x+4)}{(x+4)(x+4)}=\frac{x-5}{x+4}$$

anonymous · Answer

How? The middle term on the top doesn't have a coefficient. If it did, then that's what you would do since you would be finding the product of the last term that also made a sum equal to the middle term. There is no middle term though, it's just x.

anonymous · Answer

Yes, the middle term has a coefficient. It is 1. Because 1*x=x. That is why I replaced 1 in the quadratic formula.

anonymous · Answer

Alright, I understand now. Thank you. (:

anonymous · Answer

Are you sure? I can provide more explanation if you need? Don't hesitate please!