To determine the nature of the roots of the given equation [tex]\( x = \frac{4 \pm \sqrt{25 - 5k}}{2} \)[/tex] when [tex]\( k = 5 \)[/tex], follow these steps:
1. Substitute the value of [tex]\( k = 5 \)[/tex] into the equation:
[tex]\( x = \frac{4 \pm \sqrt{25 - 5 \cdot 5}}{2} \)[/tex].
2. Simplify the expression inside the square root:
[tex]\[ 25 - 5 \cdot 5 = 25 - 25 = 0 \][/tex].
Thus, the equation simplifies to:
[tex]\[ x = \frac{4 \pm \sqrt{0}}{2} \][/tex].
3. Evaluate the square root:
Since [tex]\(\sqrt{0} = 0\)[/tex], the equation further simplifies to:
[tex]\[ x = \frac{4 \pm 0}{2} \][/tex].
4. Write down the possible values for [tex]\( x \)[/tex]:
[tex]\[ x = \frac{4}{2} \][/tex].
Therefore, [tex]\( x = 2 \)[/tex].
5. Determine the nature of the roots:
Since the equation [tex]\( x = 2 \)[/tex] provides a single value, the roots are real and equal.
So, the nature of the roots when [tex]\( k = 5 \)[/tex] is:
a. Roots are real and equal
This can be confirmed as the true answer for the question.
