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31 December, 20:27

Find the sum of the constants a, h, and k such that

2x^2 - 8x + 7 = a (x - h) ^2 + k

for all real numbers x.

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Answers (1)
  1. 31 December, 22:46
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    3

    Step-by-step explanation:

    The value of "a" is the coefficient of x^2, so we know that is 2.

    __

    Solve for h

    Now, we have ...

    2x^2 - 8x + 7 = 2 (x - h) ^2 + k

    Expanding the right side gives us ...

    = 2 (x^2 - 2hx + h^2) + k

    = 2x^2 - 4hx + 2h^2 + k

    Comparing x-terms, we see ...

    -4hx = - 8x

    h = (-8x) / (-4x) = 2

    __

    Solve for k

    Now, we're left with ...

    2h^2 + k = 7 = 2 (2^2) + k = 8 + k

    Subtracting 8 we find k to be ...

    k = 7 - 8 = - 1

    __

    And the sum of constants a, h, and k is ...

    a + h + k = 2 + 2 - 1 = 3

    The sum of the constants is 3.
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