SOLUTION: The function h is defined by h:x -> 6x - x^2 for x ≥ 3 Express 6x-x^2 in the form a - (x - b)^2, where a and b are positive constants.

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Question 1200187: The function h is defined by
h:x -> 6x - x^2 for x ≥ 3
Express 6x-x^2 in the form a - (x - b)^2, where a and b are positive constants.
Answer by math_tutor2020(3817) About Me  (Show Source):
You can put this solution on YOUR website!

The format we want is a - (x - b)^2

Let's expand that out
a - (x - b)^2
a - (x - b)(x - b)
a - (x^2-bx-bx+b^2)
a - (x^2-2bx+b^2)
a - x^2+2bx-b^2
2bx-x^2+a-b^2
2bx-x^2+a-b^2

Compare that with 6x-x^2 to see that
2bx = 6x which leads to b = 3

So we go from
2bx-x^2+a-b^2
to
2*3x-x^2+a-3^2
to
6x-x^2+a-9

The a-9 at the end must be zero
a-9 = 0
a = 9

Therefore, we have:
a = 9
b = 3

Going back to
a - (x - b)^2
we then substitute
9 - (x - 3)^2
which is the final answer.

To verify, expand it out using similar steps shown above. I'll let the student do the verification.