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.

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.