Question

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


The solution to the square root equation is x

Answer 1

To solve the equation \( \sqrt{2x + 9} = x + 5 \), we will first square 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:

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

Now, rearranging the equation to set it to zero:

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

Combine like terms:

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

Now, we can factor the quadratic:

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

Setting the factor equal to zero gives:

\[
x + 4 = 0 \quad \Rightarrow \quad x = -4
\]

Now we need to check whether this solution satisfies the original equation:

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

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

Calculating the left side:

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

Calculating the right side:

\[
-4 + 5 = 1
\]

Since both sides are equal, \( x = -4 \) is indeed a valid solution.

Thus, the solution to the square root equation is:

\[
\boxed{-4}
\]