Question

the potential solutions to the radical equation \\(sqrt{a + 5} = a + 3\\) are \\(a = -4\\) and \\(a = -1\\). which statement is true about these solutions?\

• the solution \\(a = -1\\) is an extraneous solution.\
• the solution \\(a = -4\\) is an extraneous solution.\
• both \\(a = -4\\) and \\(a = -1\\) are true solutions.\
• neither \\(a = -4\\) nor \\(a = -1\\) are true solutions.

Explanation:

Step1: Test $a=-1$ in the equation

Substitute $a=-1$ into $\sqrt{a+5}=a+3$:
$\sqrt{-1+5} = -1+3$
$\sqrt{4} = 2$
$2=2$
This is a true statement, so $a=-1$ is a valid solution.

Step2: Test $a=-4$ in the equation

Substitute $a=-4$ into $\sqrt{a+5}=a+3$:
$\sqrt{-4+5} = -4+3$
$\sqrt{1} = -1$
$1=-1$
This is a false statement, so $a=-4$ is an extraneous solution.

Answer:

The solution $a = -4$ is an extraneous solution.