Accepted Answer

Ok, sure!

First, why don't we have a mini lesson on factoring.

Let's start with actual numbers (instead of letters), so you can see what is going on.

8 + 4
(of course, that is equal to 12)

We can factor out something from this expression, and it shouldn't change its value. Let's pull out a 2:
2(4 + 2)

What I did was put a 2 on the outside, and divide each of the inside numbers by 2 (since we are multiply on the outside and dividing on the inside, they cancel each other out).
Simplifying that:
2(6)
= 12, same thing as before

We could also pull out a 4:
4(2+1)
= 4(3) = 12, same thing!

Or even an 8:
8(1 + 0.5)
= 8(1.5) = 12, same thing again!

That shows the basics with numbers. Now, let's add a letter "x".

Simple example:
8x + 4

We can factor out a 4:
4(2x + 1)

Now, if you multiply the 4 by each term inside the parens, you should get back to where you started:
8x + 4

How about factoring out an "x"...
x(8 + 4/x)

Now, we had to divide each term on the inside by "x", so 8x/x = 8, and 4 divided by x = 4/x.
Again, you can check by multiplying -- you get back to where you started...


---------------------------
Now, your problem...

I'm assuming you understand how we got to this point -- if not, let me know.

800000t^2 - 320000t + 40000

To complete the square, we need something in the form:
a(t-h)^2 + k

You'll first notice that "t" does not have a coefficient in front of it, so it should be 1. Therefore, we need to factor out 800000 from the expression:
800000(t^2 - 0.4t) + 40000

All we have done here is like what we did in my simple examples above. We pulled out 800000, and divided those two terms by 800000:
-320000/800000 = -0.4

Good so far?

Now, we do the actual completion of the square. We need a constant term inside the parens, so that it becomes a perfect square.

Brief interlude about completing the square...
If you have something like:
(t^2-at), you have to add (a/2)^2 both inside and subtract it outside of the parens (so you don't change the final value)
Simple example:
(t^2-4t)
to complete the square, you add (-4/2)^2 = (-2)^2 = 4 (and then subtract that outside):
(t^2-4t+4) - 4
Now, it's a perfect square:
(t-2)^2 - 4

Back to your question...
Let's take (a/2)^2 for your case:
(-0.4/2)^2 = 0.04

So, we add that inside the parens, and subtract it so the final value doesn't change:
800000(t^2 - 0.4t + 0.04 - 0.04) + 40000

Bring the -0.04 outside the parens -- keep in mind that it is being multiplied by 800000, since it that is on the outside of the parens:
800000(t^2 - 0.4t + 0.04) + 40000 - 800000*0.04


Simplify:
800000(t-0.2)^2 + 8000

The rest follows as I posted last time.

Please let me know if you need further help on any of this... Please let me know which step is confusing, if you need more help.

Thanks,
Scott

Expert:	Scott
Pos. Feedback:	100.0 %
Accepts:	6483
Answered:	8/18/2008

MIT Graduate

College degree in math... proficient in all levels -- from algebra to calculus