91Ó°ÊÓ

Algebraic manipulation involves rearranging and simplifying algebraic expressions and equations. Here, the goal is to isolate the variable. First, we note that the given equation is \( x = \sqrt{x^{2}+3x+9} \). Eliminating the square root requires some careful steps.

When you manipulate algebraic expressions:

• Identify like terms
• Use operations like addition, subtraction, multiplication, and division

In this specific exercise, we start by recognizing the operations within the equation. Next, we prepare to use one of the most common techniques to remove the square root: squaring both sides of the equation.

Squaring both sides is a method used to eliminate square roots by raising both sides of an equation to the power of two. This step is critical for solving radical equations. Here’s how it works:Starting with \( x = \sqrt{x^{2}+3x+9} \), we square both the left and the right sides of the equation:
\( x^2 = (\sqrt{x^{2}+3x+9})^2 \)
This simplifies the equation to:
\( x^2 = x^2 + 3x + 9 \)By removing the square root, you transform the problem into a polynomial equation, which is easier to solve. However, note that squaring both sides might introduce extraneous solutions, which we will need to check later.

Validating solutions is essential to ensure that the solutions obtained are indeed correct, especially after squaring or other non-linear operations. After simplifying and solving the equation, substituting back into the original equation confirms the validity.
For instance, in this problem, we found that \( x=3 \) is the solution. Substituting \( x=3 \) back into the original equation \( x = \sqrt{x^{2}+3x+9} \) leads to:
\( -3 = \sqrt{(9 - 9 + 9)} \)
This means:\( -3 = \sqrt{(9)} \)
which simplifies to:

\( -3 = 3 \)
Since there is no way \(-3 = 3 \), we conclude that our initial solution \( x=-3 \) is not valid. Thus, the equation has no actual solution after verification.
This highlights the importance of the validation step to detect extraneous solutions often introduced by squaring both sides.