Question 117946: Hi, this question is a multiple choice question from the pre-ACT test. I don't even know where to start with this one, I tried solving for x, and I tried to put the answer choices in the problem, but to no avail. I believe this is an identity but I can't figure out how to use that knowledge to solve this. This problem has frustrated me for 2 days.
In the equation, x^2 + 16x + a = (x + b)^2, a and b are constants. If the equation is true for all values of x, what is the value of a?
the answer choices are: 4, 8, 16, 64, or 256. The correct answer is 64 but I have no idea how to work out the problem to get that answer. Please help me figure this problem out.
thanks for your help!
Found 2 solutions by Earlsdon, Edwin McCravy:
Answer by Earlsdon(6103)   (Show Source):
You can put this solution on YOUR website!Here's one approach you could try.
 Expand the right side.
 Subtract  from both sides.
 Now equate the x-terms.
 Divide by 2x.
Now equate the constant terms.
 but b = 8, so...
 and...
Answer by Edwin McCravy(6938)  (Show Source):
You can put this solution on YOUR website!
x² + 16x + a = (x + b)²
x² + 16x + a = (x + b)(x + b)
x² + 16x + a = x² + bx + bx + b²
x² + 16x + a = x² + 2bx + b²
Since the left and right sides must be
equal for all values of x, the corresponding
terms on each side must be equal for all x.
So we have
x² = x²
16x = 2bx
a = b²
The second equation tells us 16 = 2b or b = 8
Substituting b = 8 into the third equation
gives
a = b²
a = 8²
a = 64
Edwin
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