| MWUHAHAH I saved the best for last! loganr typed problem 24 wrong. Here it is:
24) If n > 0 and 9x^2 + kx + 36 = (3x+2)<b>^2</b> for all values of x, what is the value of k - n?
Hahahahah this was awesome. Since I found the mistake, I get to solve it, woohoo!
You know that (3x + n) = 9x^2 + 6xn + n^2, right? Doesn't that look oddly familiar? Ah, yes! The left side of the equation! But we'll get back to that in a minute.
On the left side, aren't the 9 and 36 perfect squares? Hm.. perhaps we'll be able to use that. Can we factor? Hey, since the right side is a perfect square, can the left side be one, too? Using just the 9 and the 36, the only possible square would be (3x + 6)^2. When expanded, that equals 9x^2 + 12x + 36. IF that works, then k = 36. Now, over to the right side. Hey, that (3x + n) looks similar to the (3x + 6) we squared a minute ago. Since both sides are equal, n would have to equal 6. Now, expand both sides, plugging in 36 for k and 6 for n. My word, it works! k - n = 36 - 6 = 30 --> D.
I'm not sure why I narrate in math problems. You'll just have to suck it up and deal :P.
The 9x^2 easily drop out; we don't need to look at them anymore. Now, the problem says FOR ALL VALUES OF X, so there must be more than one possible value of x. Therefore, we cannot expect it to have only one value. To accomodate this |