Sure, let's solve the equation [tex]\(\sqrt{x^2 + 49} = x + 5\)[/tex] step by step.
1. Isolate the square root:
[tex]\[
\sqrt{x^2 + 49} = x + 5
\][/tex]
2. Square both sides to eliminate the square root:
[tex]\[
\left(\sqrt{x^2 + 49}\right)^2 = (x + 5)^2
\][/tex]
[tex]\[
x^2 + 49 = x^2 + 10x + 25
\][/tex]
3. Simplify the equation:
Subtract [tex]\(x^2\)[/tex] from both sides:
[tex]\[
x^2 + 49 - x^2 = x^2 + 10x + 25 - x^2
\][/tex]
[tex]\[
49 = 10x + 25
\][/tex]
4. Solve for [tex]\(x\)[/tex]:
Subtract 25 from both sides:
[tex]\[
49 - 25 = 10x
\][/tex]
[tex]\[
24 = 10x
\][/tex]
Divide both sides by 10:
[tex]\[
x = \frac{24}{10}
\][/tex]
Simplify the fraction:
[tex]\[
x = \frac{12}{5}
\][/tex]
So, the solution to the equation [tex]\(\sqrt{x^2 + 49} = x + 5\)[/tex] is:
[tex]\[
x = \frac{12}{5}
\][/tex]
Let's verify the solution to ensure it's correct.
Substitute [tex]\(x = \frac{12}{5}\)[/tex] back into the original equation:
[tex]\[
\sqrt{\left(\frac{12}{5}\right)^2 + 49} = \frac{12}{5} + 5
\][/tex]
[tex]\[
\sqrt{\frac{144}{25} + 49} = \frac{12}{5} + \frac{25}{5}
\][/tex]
[tex]\[
\sqrt{\frac{144 + 1225}{25}} = \frac{37}{5}
\][/tex]
[tex]\[
\sqrt{\frac{1369}{25}} = \frac{37}{5}
\][/tex]
[tex]\[
\frac{37}{5} = \frac{37}{5}
\][/tex]
This confirms that [tex]\(x = \frac{12}{5}\)[/tex] is indeed the correct solution. Thus, the answer is:
[tex]\[
x = \frac{12}{5}
\][/tex]
