Graphing and solving Quadratic Inequalities?

[Goldenboko]

I need some help. I want to move on to Pre-Calculus Honors next year but in order to I have to learn extra stuff on my own, and pass a test. I'm doing okay but...

While checking an example problem's answer, I found something that I can't make sense of.

Here is what the question is.

A rectangle is 6 centimeters longer than it is wide. Find the dimensions if the area of the rectangle is more then 216sqcm.

I did it out, and went to check.

Here's what the packet said:

1. w(w+6)>216
2. w2 (that two means squared) + 6w> 216
3. w2+62-216>0
4. (w2 + 6w + 9) -216 -9>0
5. (w+3)2 (that two means squared) -225>0

Now its step 4 that makes me go o.0

I understand why they got nine, they're trying to complete the square...

BUT!

I don't understand why they are subtracting the nine. In order to get the last number in a completed square you do (b/2)2, so how did they get a negative nine? Isn't any number squared positive?

Help plz Dx

[rubah]

If you add something you have to subtract it else where, so that it remains identically equal. Sometimes they would just set it equal to +9, but here they chose to subtract it.

[edit- haha, i finished it out from there, and I'm positively tickled because I figured I couldn't do that stuff anymore xD]

[Tavrobel]

I disagree with the packet's method. Unless it tells you to do something specifically, use a different way. I still don't use CTS whenever possible. Also, you can represent exponents on the keyboard using Shift+6, the carrot symbol.

In case you didn't get the answer:
Draw a picture. Then copy down the problem.
x*(6 + x) <= 216

Distribute. And bring over the 216, so that way you have something you should be familiar with, a polynomial.
x² + 6x - 216 <= 0

Either complete the square (which is why they are getting a nine), or use the quadratic formula. Substitute coefficients.
(-b +/- Radical(b² - 4ac)) / 2a
(-6) +/- Radical( (36) - (4)(1)(-216) ) / 2
(-6) +/- (900)^1/2 / 2
(-3) +/- (15)
== -18 or 12.

Substitute -18 and 12.
Whichever one gives you a valid solution will be correct.
(-18)² + 6(-18) - 216 <= 0
It's less than zero, but you'll get a negative distance.
(12)² + 6(12) - 216 <= 0
== 0.

You're not done. Just because x == 12 doesn't mean you know the area. So, substitute.
(6+12)(12) = 18 (length) and 12 (width) in centimeters.

[Goldenboko]

In this case I would be done once finding the correct inequality, as the packet is asking me to graph the parabola.

Anyway, thanks, this probably won't be the last time I ask a question about this packet xD

[Jessweeee♪]

Aren't quadratic equations the funnest :D?

[Goldenboko]

Aren't quadratic equations the funnest ?

Indeed.