To solve the equation [tex]\((x + 49)^2 = 24\)[/tex], we need to isolate [tex]\(x\)[/tex]. Here's a step-by-step solution:
1. Start with the given equation:
[tex]\[
(x + 49)^2 = 24
\][/tex]
2. Take the square root of both sides to undo the square on the left-hand side. Remember, when taking the square root, you need to consider both the positive and the negative roots:
[tex]\[
x + 49 = \pm \sqrt{24}
\][/tex]
3. Simplify the square root of 24. Note that [tex]\(\sqrt{24} = \sqrt{4 \times 6} = 2\sqrt{6}\)[/tex], so:
[tex]\[
x + 49 = \pm 2\sqrt{6}
\][/tex]
4. Solve for [tex]\(x\)[/tex] by isolating it on one side of the equation:
[tex]\[
x = -49 \pm 2\sqrt{6}
\][/tex]
Thus, the solutions to the equation [tex]\((x + 49)^2 = 24\)[/tex] are:
[tex]\[
x = -49 + 2\sqrt{6} \quad \text{and} \quad x = -49 - 2\sqrt{6}
\][/tex]
This fits with the provided multiple-choice options, where the correct answer is:
[tex]\[
x=-49 \pm 2 \sqrt{6}
\][/tex]
