To determine the value(s) of [tex]\( k \)[/tex] for which the roots of the quadratic equation [tex]\( x = \frac{4 \pm \sqrt{25 - 5k}}{2} \)[/tex] are non-real, we need to look at the discriminant of the equation inside the square root.
The expression inside the square root is [tex]\( 25 - 5k \)[/tex]. For the roots to be non-real, this expression must be negative because the square root of a negative number results in an imaginary number (i.e., non-real).
1. We start by setting up the inequality:
[tex]\[
25 - 5k < 0
\][/tex]
2. Isolate [tex]\( k \)[/tex] by solving the inequality:
[tex]\[
25 < 5k
\][/tex]
3. Divide both sides by 5 to solve for [tex]\( k \)[/tex]:
[tex]\[
5 < k
\][/tex]
Therefore, the roots will be non-real if:
[tex]\[
k > 5
\][/tex]
The correct answer is:
b. [tex]\( k > 5 \)[/tex]
