Given: [tex] x=\frac{4 \pm \sqrt{25-5k}}{2} [/tex]

If [tex] k=5 [/tex], determine the nature of the roots:

A. Roots are real and equal
B. Roots are non-real and equal
C. Roots are real and unequal
D. Roots are non-real and unequal



Answer :

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.