Graphing and solving Quadratic Inequalities?

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

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]

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.

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

Aren't quadratic equations the funnest :D?

Quote:

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Originally Posted by Jessweeee♪

Aren't quadratic equations the funnest :D?

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Indeed.