Certainly! Let's solve the quadratic inequality [tex]\(4(x+2)^2 \leq 0\)[/tex] step by step.
1. Understand the inequality: You are given the quadratic inequality [tex]\(4(x + 2)^2 \leq 0\)[/tex].
2. Solve the corresponding equation: To solve this inequality, we first solve the corresponding equation [tex]\(4(x + 2)^2 = 0\)[/tex].
3. Simplify the equation:
[tex]\[ 4(x + 2)^2 = 0 \][/tex]
[tex]\[ (x + 2)^2 = 0 \][/tex]
4. Solve for [tex]\(x\)[/tex]:
[tex]\[ x + 2 = 0 \][/tex]
[tex]\[ x = -2 \][/tex]
5. Analyze the inequality:
The expression [tex]\((x+2)^2\)[/tex] represents a squared term, which is always non-negative. Therefore, [tex]\((x+2)^2\)[/tex] is zero when [tex]\(x = -2\)[/tex].
So, [tex]\(4(x + 2)^2\)[/tex] is zero when [tex]\(x = -2\)[/tex] and is positive for all other values of [tex]\(x\)[/tex].
6. Determine the inequality: Since [tex]\(4(x + 2)^2 \leq 0\)[/tex] implies that the expression is less than or equal to zero, this can only be true when [tex]\(4(x + 2)^2 = 0\)[/tex]. As calculated earlier, this happens only when [tex]\(x = -2\)[/tex].
7. Conclusion: The inequality [tex]\(4(x + 2)^2 \leq 0\)[/tex] holds true only when [tex]\(x = -2\)[/tex].
Therefore, the solution set for the quadratic inequality [tex]\(4(x + 2)^2 \leq 0\)[/tex] is:
[tex]\[ \{x \mid x = -2\} \][/tex]
Answer: [tex]\(\{x \mid x = -2\}\)[/tex]
