Certainly! Let's solve the equation [tex]\(\frac{x^2 + 5}{x + 5} = \frac{30}{x + 5}\)[/tex] step-by-step and eliminate any extraneous solutions.
1. Start with the given equation:
[tex]\[
\frac{x^2 + 5}{x + 5} = \frac{30}{x + 5}
\][/tex]
2. Since both sides of the equation have the common denominator [tex]\((x + 5)\)[/tex], first consider the scenario where [tex]\(x \neq -5\)[/tex] to avoid division by zero:
[tex]\[
(x^2 + 5) = 30
\][/tex]
3. Solving the simplified equation:
[tex]\[
x^2 + 5 = 30
\][/tex]
Subtract 5 from both sides:
[tex]\[
x^2 = 25
\][/tex]
4. Take the square root of both sides:
[tex]\[
x = \pm 5
\][/tex]
Hence, the potential solutions are [tex]\(x = 5\)[/tex] and [tex]\(x = -5\)[/tex].
5. Check for extraneous solutions:
Since we earlier excluded [tex]\(x = -5\)[/tex] to avoid division by zero (since [tex]\(\frac{x^2 + 5}{x + 5}\)[/tex] and [tex]\(\frac{30}{x + 5}\)[/tex] would be undefined at [tex]\(x = -5\)[/tex]), we discard [tex]\(x = -5\)[/tex] as an invalid solution.
6. Confirm the valid solution:
We are left with [tex]\(x = 5\)[/tex].
So, the only valid solution to the equation is:
[tex]\[
x = 5
\][/tex]
