Let's solve this step-by-step.
Given the quadratic equation [tex]\( x^2 + bx + 16 = 0 \)[/tex], we are told that the only solution to this equation is [tex]\( x = 4 \)[/tex].
A quadratic equation can have at most two solutions. If [tex]\( x = 4 \)[/tex] is the only solution, it means the equation has a repeated root, which makes it a perfect square. Therefore, the equation can be written in the form:
[tex]\[ (x - 4)^2 = 0 \][/tex]
Expanding this, we get:
[tex]\[ x^2 - 8x + 16 = 0 \][/tex]
Now, let's compare this with the original given equation [tex]\( x^2 + bx + 16 = 0 \)[/tex].
By comparing the coefficients of [tex]\( x \)[/tex], we see that:
[tex]\[ -8 = b \][/tex]
So, the value of [tex]\( b \)[/tex] is:
[tex]\[ b = -8 \][/tex]
Therefore, the correct answer is:
[tex]\[ b = -8 \][/tex]
