To solve the equation [tex]\((x-4)^2 + 4 = 49\)[/tex], let's go through the solution step-by-step:
1. Simplify the equation:
Start by rewriting the given equation:
[tex]\[
(x-4)^2 + 4 = 49
\][/tex]
2. Isolate [tex]\((x-4)^2\)[/tex]:
Subtract 4 from both sides to isolate the squared term:
[tex]\[
(x-4)^2 + 4 - 4 = 49 - 4
\][/tex]
[tex]\[
(x-4)^2 = 45
\][/tex]
3. Solve for [tex]\(x-4\)[/tex]:
To solve for [tex]\(x-4\)[/tex], take the square root of both sides:
[tex]\[
\sqrt{(x-4)^2} = \sqrt{45}
\][/tex]
Remember that taking the square root gives both positive and negative solutions:
[tex]\[
x-4 = \pm \sqrt{45}
\][/tex]
4. Simplify [tex]\(\sqrt{45}\)[/tex]:
Note that [tex]\(\sqrt{45}\)[/tex] can be simplified:
[tex]\[
\sqrt{45} = \sqrt{9 \times 5} = \sqrt{9} \times \sqrt{5} = 3\sqrt{5}
\][/tex]
Therefore, we have:
[tex]\[
x-4 = \pm 3\sqrt{5}
\][/tex]
5. Solve for [tex]\(x\)[/tex]:
Isolate [tex]\(x\)[/tex] by adding 4 to both sides:
[tex]\[
x = 4 \pm 3\sqrt{5}
\][/tex]
6. Conclude the solutions:
The solutions to the equation [tex]\((x-4)^2 + 4 = 49\)[/tex] are:
[tex]\[
x = 4 - 3\sqrt{5} \quad \text{and} \quad x = 4 + 3\sqrt{5}
\][/tex]
Hence, the solutions are:
[tex]\[ x = 4 - 3\sqrt{5} \quad \text{and} \quad x = 4 + 3\sqrt{5} \][/tex]
None of the other options provided ([tex]\(x= -4 \pm 2 \sqrt{10}\)[/tex], [tex]\(x= \pm 7\)[/tex], [tex]\(x=-4 + \sqrt{45}\)[/tex]) match our found solutions. The correct solution is indeed:
[tex]\[ x = 4 - 3\sqrt{5} \quad \text{and} \quad x = 4 + 3\sqrt{5} \][/tex]
