If the real solution to the equation |x^2 + bx + 16| = 8 is 16 what is the value of b.

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The real solution to the equation |x^2 + bx + 16| = 8 is 16.

|x^2 + bx + 16| = 8 gives two equations for each of which the values of b can be determined.

The two equations are x^2 + bx + 16 = 8 and x^2 + bx + 16 = -8

=> x^2 + bx + 8 = 0 and x^2 + bx + 24 = 0

The roots of x^2 + bx + 8 = 0 are:

`(-b + sqrt(b^2 - 32))/2` and `(-b - sqrt(b^2 - 32))/2`

`(-b + sqrt(b^2 - 32))/2 = 16`

=> `-b + sqrt(b^2 - 32) = 32`

=> `(32 + b)^2 = b^2 - 32`

=> `1024 + b^2 + 64b = b^2 - 32`

=> b = -16.5

The same value of b is obtained for the other root of the equation.

For the equation x^2 + bx + 24 = 0

`(-b + sqrt(b^2 - 96))/2 = 16`

=> `-b + sqrt(b^2 - 96) = 32`

=> `b^2 - 96 = 1024 + b^2 + 64b`

=> `64b = -1120`

=> b = -17.5

**The values of b that satisfy the given condition are -16.5 and -17.5**

Another way to do this is to substitute x=16 into the equation.

`|16^2+b(16)+16|=8` and we get

`|256+16b+16| = 8` simplifying we get

`|16b+272|=8`

Now we can split the equation to get

`16b+272=8` or `16b+272=-8`

Subtracting 272 from both equations we get

`16b=-264` or `16b=-280`

Dividing by 16 we get

`b = -264/16 = -16.5` or `b=-280/16 = -17.5`

So our answer is b=-17.5 or b=-16.5

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