Solve \$\sqrt(m + 2) + 1 = \sqrt(3 - m)\$.

Write the original equation:

\$\sqrt(m + 2) + 1 = \sqrt(3 - m)\$

Square each side

\$(\sqrt(m + 2) + 1)^2 = (\sqrt(3 - m))^2\$

Expand left side and simplify right side

\$m + 2 + 2\sqrt(m + 2) + 1 = 3 - m\$

Isolate radical expression

\$2\sqrt(m + 2) = -2m\$

Divide each side by 2

\$\sqrt(m + 2) = -m\$

Square each side

\$(\sqrt(m + 2))^2 = (-m)^2\$

Simplify

\$(\sqrt(m + 2))^2 = (-m)^2\$

\$m + 2 = m^2\$

Write in standard form and factorize

\$- m^2 + m + 2 = 0\$

\$(m + 1)(m - 2) = 0\$

m = -1 and m = 2

Check both solutions in the original equation to ensure they are valid: For m = -1:

\$\sqrt(m + 2) + 1 = \sqrt(3 - m)\$

\$\sqrt(-1 + 2) + 1 = \sqrt(3 - (-1))\$

\$\sqrt(1) + 1 = \sqrt(4)\$

2 = 2

For m = 2:

\$\sqrt(m + 2) + 1 = \sqrt(3 - m)\$

\$\sqrt(2 + 2) + 1 = \sqrt(3 - 2)\$

\$\sqrt(4) + 1 = \sqrt(1)\$

\$3 ne 1\$

The apparent solution m = 2 is extraneous. So, the only solution is m = -1.

This value satisfies the equation as substituting m=−1 makes both sides equal

m = -1

This value does not satisfy the equation as substituting m=−2 results in unequal sides

m = -2

This value does not solve the equation correctly, as substituting m=1 yields different values for both sides

m = 1

This value fails to satisfy the equation because substituting m=2 does not make both sides equal

m = 2

Option

A

Can you summarize what you’ve understood in the above steps?