Question

Solve the square root equation
√
2
x
+
9
=
x
+
5
.(1 point)


The solution to the square root equation is
x

=

Answer 1

To solve the square root equation

\[
\sqrt{2x + 9} = x + 5,
\]

we start by squaring both sides to eliminate the square root:

\[
(\sqrt{2x + 9})^2 = (x + 5)^2.
\]

This simplifies to:

\[
2x + 9 = (x + 5)(x + 5).
\]

Expanding the right side gives:

\[
2x + 9 = x^2 + 10x + 25.
\]

Next, we rearrange the equation to set it to zero:

\[
0 = x^2 + 10x + 25 - 2x - 9.
\]

This simplifies to:

\[
0 = x^2 + 8x + 16.
\]

Now, we factor the quadratic:

\[
0 = (x + 4)(x + 4).
\]

This gives us:

\[
0 = (x + 4)^2.
\]

Thus, we find the solution:

\[
x + 4 = 0 \implies x = -4.
\]

Next, we need to check if this solution satisfies the original equation:

Substituting \(x = -4\) into the original equation:

\[
\sqrt{2(-4) + 9} = -4 + 5.
\]

Calculating the left side:

\[
\sqrt{-8 + 9} = \sqrt{1} = 1,
\]

and the right side:

\[
-4 + 5 = 1.
\]

Since both sides are equal, we have confirmed that the solution is correct. Therefore, the solution to the square root equation is:

\[
\boxed{-4}.
\]