Problem 26 - Algebra A Combined Function

Question

Solve. $$ -\sqrt{3 x+9}=-12 $$

Step-by-Step Solution

Answer

The solution is \(x = 45\).

Step 1: Isolate the Square Root

First, we need to isolate the square root on one side of the equation. The equation is already given as \(-\sqrt{3x + 9} = -12\). Since both sides are negative, we can multiply both sides by -1 to remove the negatives, giving us \(\sqrt{3x + 9} = 12\).

Step 2: Square Both Sides

To eliminate the square root, square both sides of the equation: \((\sqrt{3x + 9})^2 = 12^2 \)This results in \(3x + 9 = 144.\)

Step 3: Solve for x

Subtract 9 from both sides to isolate the term with \(x\):\(3x = 144 - 9\)Simplifying this gives:\(3x = 135.\)Next, divide both sides by 3 to solve for \(x\):\(x = \frac{135}{3}.\)Thus, \(x = 45\).

Step 4: Check the Solution

It's important to verify the solution by substituting back into the original equation. Substituting \(x = 45\) into \(\sqrt{3x + 9}\), we get \(\sqrt{3(45) + 9} = \sqrt{135 + 9} = \sqrt{144} = 12.\)Since \(-12 = -12\) holds true, our solution is verified as correct.

Key Concepts

Square Root Isolation

In algebra, isolating the square root is often a critical step in solving equations involving a square root. In this problem, you start with the equation \(-\sqrt{3x + 9} = -12\).Here, the square root is already isolated on one side, thanks to the negative signs balancing each other out. You can simplify things by multiplying both sides by \(-1\) to eliminate the negatives. This leads to:\[ \sqrt{3x + 9} = 12 \]Now the square root is isolated, making it easier to eliminate in the next steps.
Be patient with this process. It's an essential step to simplify the problem and move toward finding the solution.

Solving Quadratic Equations

With an isolated square root, the next step is to eliminate the square root to deal with a simpler equation. Here, squaring both sides of the equation is the way to do this:\( (\sqrt{3x + 9})^2 = 12^2 \).Squaring the square root cancels it out, leaving you with the expression:\[ 3x + 9 = 144 \]Now, solve for \(x\) by isolating it. Subtract 9 from both sides:\[ 3x = 135 \]Then, divide by 3:\[ x = \frac{135}{3} \]Which simplifies to:\[ x = 45 \]This process transforms the initial equation into a straightforward arithmetic problem, allowing you to determine the value of \(x\).

Verifying Solutions

Once you find a potential solution, it's crucial to verify that it satisfies the original equation. Substitute the solution \(x = 45\) back into the equation to check your work. You replace \(x\) in the expression \(\sqrt{3x + 9}\) as follows:\( \sqrt{3(45) + 9} = \sqrt{135 + 9} = \sqrt{144} \)This simplifies down to \(12\), which matches the right side of the adjusted equation.This means \(-12 = -12\), confirming that the solution is correct.
Always verify your solutions as this process ensures that the answer logically fits within the constraints of the original equation, catching any potential errors made along the way.