Test 9 Section 4 Question 16

(4x+4)(ax-1)-x^2+4

This question says that the whole expression above resolves itself to just equal bx. Which means all the constant terms and x^2 terms need to cancel out somehow. We can use that fact to figure out what a is.

First, let’s FOIL it out:

\left(4ax^2-4x+4ax-4\right)-x^2+4

Now let’s combine like terms:

4ax^2-x^2-4x+4ax-4+4

Again, this is supposed to simplify to just bx. Obviously the -4 and the +4 cancel out, and the 4ax^2-x^2 must also cancel out.

4ax^2-x^2=0

Divide each side by x^2 and you get simply:

4a-1=0\\4a=1\\\\a=\dfrac{1}{4}

Now that we know a, we can solve for b.

-4x+4ax\\\\=-4x+4\left(\dfrac{1}{4}\right)x\\\\=-4x+x\\=-3x

Therefore, b=-3.

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