Question

|x^{2}-16x+56| = -4x-7

what does x equal?

Answer 1

If the | ...| means absolute value, then

x^2 - 16x + 56 = -4x-7
or
-x^2 + 16x - 56 = -4x - 7

case1:
x^2 - 12x + 63 = 0
complete the square:
x^2 - 12x + 36 = -63 + 36
(x-6)^2 = -27
no real solution, I have not seen complex numbers associated with absolute value

case2:
-x^2 + 20x - 49 = 0
x^2 - 20x + 49 = 0
x^2 - 20x + 100 = -49+100
(x-10)^2 = 51
x-10 = ±√51
x = 10±√51
x = appr 17.14 or appr 2.86

but we also know that -4x-7 ≥0
-4x≥7
x ≤ -7/4

so none of the anwers work,
no solution