Alan

The difference is given by p(x² - (x-2)²) + q(x - (x-2)) or (4x-4)p + 2q

For this to equal (2x - 1)2 we must have 4p = 4 or p= 1 and 2q - 4p = -2, so 2q - 4 = -2 or q = 1

p+q = 2

"If A its nxn Matrix and A^4 - A^3 + A^2 - A + Ιnxn = 0nxn, prove Α^-1 = -Α^4"

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"What is the value of k if point P(k, k sqrt(3)) is 4 units from the line 5x+3y=12?"

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Presumably, you want to know the values of each of the angles.  If so:

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Drawing the triangle might help:

(and, no, it's not "adjacent").

In general lg(u^a) = a*lg(u).  When a is -1 then lg(u^-1) = -1*lg(u)  or  lg(1/u) = -lg(u)

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Here's my take:

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As follows:

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You can do matrix calculations on the home page calculator here.  See the following example:

You might be making this more complicated than necessary!

As \(x\rightarrow -\infty\) the 2 in the numerator is negligible compared with 9x², and the -9 in the denominator is negligible compared with the 2x.  So:

\(\lim_{x\rightarrow -\infty}\frac{\sqrt{9x^2+2}}{2x-9}\rightarrow\lim_{x\rightarrow -\infty}\frac{\sqrt{9x^2}}{2x}\rightarrow \lim_{x\rightarrow -\infty}\frac{3|x|}{2x}\)

Now, when x is negative \(\frac{|x|}{x}=-1\) so the limit is -3/2.