Problem 59 - Algebra 2

Question

Solve \(\sqrt{4 x-23}-3=2\)

Step-by-Step Solution

Answer

The solution to the equation \(\sqrt{4 x-23}-3=2\) is \(x = 12\).

Step 1: Isolate the square-root term

To start off, the square-root term should be isolated. Therefore, add 3 on both sides of the equation to get rid of the -3 from the left-hand side. So, \(\sqrt{4 x-23}-3+3=2+3\) giving us \(\sqrt{4 x-23}=5\).

Step 2: Remove the square-root

Now, let's get rid of the square root on the left-hand side. To do this, square both sides of the equation. So \((\sqrt{4 x-23})^2 = 5^2\) which simplifies to \(4x-23=25\).

Step 3: Isolate \(x\)

Now that we have a linear equation, we can easily solve for \(x\). First, add 23 on both sides of the equation to get rid of the -23 on the left. It gives \(4x = 25 + 23\) simplifying to \(4x = 48\). Finally, divide both sides of the equation with 4 to isolate \(x\). So \(x = \frac{48}{4}\) gives \(x = 12\) as the solution.

Key Concepts

Isolating Radical Expressions

When faced with an equation that includes a square-root term, such as \( \sqrt{4x-23} - 3 = 2 \), the first step is to isolate the radical expression on one side of the equation. This involves rearranging the equation so that the square-root term stands alone. Here's how it's done:

• Identify any terms that are being added or subtracted from the square-root expression. In your example, this is \(-3\).
• Use addition or subtraction to eliminate these terms. Add \(3\) to both sides in this case, resulting in \( \sqrt{4x-23} = 5 \).

By isolating the radical, you're setting up the equation for the next important step. Always double-check your arithmetic to ensure that only the square-root term remains on one side. This process is crucial because it simplifies solving the equation in the next steps, especially when dealing with more complex radical equations.

Squaring Both Sides

With the radical expression isolated, the next step is to eliminate the square root itself. You accomplish this by squaring both sides of the equation. This action will help you transform the expression into a more manageable form. Remember, whatever operation you do to one side of an equation, you must do to the other to maintain equality.Let's break that down:

• Square the radical term. For \( \sqrt{4x-23} = 5 \), squaring \( \sqrt{4x-23} \) removes the square root: \( (\sqrt{4x-23})^2 = 4x-23 \).
• Square the opposite side of the equation as well: \( 5^2 = 25 \).

Now, you have a linear equation: \( 4x - 23 = 25 \). Squaring both sides is a powerful technique, but remember that it can potentially introduce extraneous solutions. Always verify your final answers in the original equation to ensure they hold true.

Solving Linear Equations

Once the radical has been removed, you typically end up with a linear equation, ready for solving. Linear equations are straightforward but require careful handling to isolate the variable completely. Here, the equation is \( 4x - 23 = 25 \).Proceed through these steps:

• Add \(23\) to each side. This step eliminates the constant term on the left: \( 4x = 25 + 23 \), simplifying to \( 4x = 48 \).
• Divide both sides by \(4\) to solve for \(x\): \( x = \frac{48}{4} \), giving you \( x = 12 \).

Solving a linear equation involves basic arithmetic — utilizing addition, subtraction, multiplication, or division to isolate the variable. After finding \(x\), always substitute back into the original equation to confirm that the solution satisfies it. This ensures that there are no errors or extraneous solutions arising from previous steps.