Help explaining equation simplifying

Question

here's what I'm working with:

pi ([(24-2s)/pi]^2) + s^2 = [(4/pi) + 1]* s^2 - 96s/pi + 576/pi

i have no idea where the (4/pi + 1) came from can someone walk me through the simplifying?

Answer 1

Hello Ayato,

The struggle with explaining this simplification is that typing the notation is not easy . . . but we will give it a try!

pi [(24-2s)/pi]^2 + s^2 is where we start. I am going to square the (24-2s)/pi

This gives [(576 - 96s + 4s^2]/pi^2. When I multiply this by the pi that is in front of the original expression, one of pi's cancels in the bottom leaving us with

[(576 - 96s + 4s^2]/pi + s^2 (I suspect you got this far!)

To add these two terms, I need a common denominator. The denominator of the first term is pi. The s^2 has a denominator of 1 so I will mulitply it by pi/pi. This gives

[(576 - 96s + 4s^2]/pi + pi*s^2/pi

I can them add the numerators and place them over the common denominator, giving us

(576 - 96s + 4s^2 + pi*s^2)/pi

The last two terms in the numerator both have an s^2, so it can be factored out of those two terms leaving us with

(576 - 96s + (4+pi)*s^2)/pi

Now we divide each term by pi since it is in the denominator . . .

576/pi - 96s/pi + (4 + pi)s^2/pi

We are close!!!

(4 + pi)s^2/pi can be regrouped as [(4 + pi)/pi]*s^2 which simplifies to

[4/pi + pi/pi] * s^2 which simplifies again to [4/pi + 1]*s^2

This finishes the simplication and you have

576/pi - 96s/pi + [4/pi + 1]*s^2

I hope this helps!

Jim