To determine the solution set of the quadratic inequality \(x^2 - 5 \leq 0\), we need to solve for \(x\) by following these steps:
1. Rewrite the inequality:
[tex]\[ x^2 - 5 \leq 0 \][/tex]
2. Bring the inequality to standard form:
[tex]\[ x^2 \leq 5 \][/tex]
3. Isolate \(x\) by taking the square root of both sides:
[tex]\[ |x| \leq \sqrt{5} \][/tex]
This means \(x\) can be any value between \(-\sqrt{5}\) and \(\sqrt{5}\).
4. Express the solution as an interval:
The absolute value inequality \( |x| \leq \sqrt{5} \) translates to:
[tex]\[ -\sqrt{5} \leq x \leq \sqrt{5} \][/tex]
Thus, the solution set of the quadratic inequality \(x^2 - 5 \leq 0\) is:
[tex]\[ \{x \mid -\sqrt{5} \leq x \leq \sqrt{5}\} \][/tex]
Therefore, the correct answer is:
[tex]\[ \{x \mid -\sqrt{5} \leq x \leq \sqrt{5}\} \][/tex]
