Solve the equation, eliminating any extraneous solutions.

[tex]\[
\frac{x^2+5}{(x+5)}=\frac{3}{(x+5)}
\][/tex]

A. [tex]\(x=-5\)[/tex]
B. [tex]\(x=-5, 5\)[/tex]
C. [tex]\(x=5\)[/tex]
D. [tex]\(x=-25\)[/tex]



Answer :

To solve the equation

[tex]\[ \frac{x^2 + 5}{x + 5} = \frac{3}{x + 5} \][/tex]

we follow these steps:

1. Identify Common Denominator:
Since both sides of the equation have the same denominator [tex]\(x + 5\)[/tex], we can equate the numerators directly (provided [tex]\(x \neq -5\)[/tex] to avoid division by zero).

2. Equate the Numerators:
Set the numerators equal to each other:

[tex]\[ x^2 + 5 = 3 \][/tex]

3. Solve for [tex]\(x\)[/tex]:
Rearrange the equation to solve for [tex]\(x\)[/tex]:

[tex]\[ x^2 + 5 = 3 \][/tex]

Subtract 5 from both sides:

[tex]\[ x^2 = 3 - 5 \][/tex]

Simplify the right-hand side:

[tex]\[ x^2 = -2 \][/tex]

4. Consider the Result:
Taking the square root of both sides, we get:

[tex]\[ x = \sqrt{-2} \][/tex]

The square root of a negative number is not a real number. Therefore, [tex]\(x = \sqrt{-2}\)[/tex] does not produce any real solutions.

Since the problem requires real solutions, we conclude that:

[tex]\[ \boxed{\text{There are no real solutions.}} \][/tex]