Problem 14 - Algebra for College Students

Question

Solve each equation. Don't forget to check each of your potential solutions. \(\sqrt{4 x-1}-3=2\)

Step-by-Step Solution

Answer

The solution is \(x = 6.5\).

Step 1: Isolate the square root

Start by isolating the square root in the equation. Add 3 to both sides of the equation: \[ \sqrt{4x-1} = 2 + 3 \] which simplifies to \[ \sqrt{4x-1} = 5 \].

Step 2: Eliminate the square root

Eliminate the square root by squaring both sides of the equation: \[ (\sqrt{4x-1})^2 = 5^2 \]which simplifies to \[ 4x - 1 = 25 \].

Step 3: Solve for x

Now that the square root is eliminated, solve for \(x\):Add 1 to both sides:\[ 4x = 26 \]Divide both sides by 4:\[ x = \frac{26}{4} \]Simplifying the fraction gives:\[ x = \frac{13}{2} \] or \(x = 6.5\).

Step 4: Check the solution

Substitute \(x = 6.5\) back into the original equation to verify the solution:\[ \sqrt{4(6.5) - 1} - 3 = 2 \]Calculate \(4(6.5) - 1\):\[ 4 \times 6.5 = 26 \]\[ 26 - 1 = 25 \]\[ \sqrt{25} - 3 = 2 \]Calculate \(\sqrt{25}\):\[ 5 - 3 = 2 \]The original equation holds true, so \(x = 6.5\) is the correct solution.

Key Concepts

Isolating the square root

In solving square root equations, the first step is always to isolate the square root expression. This means you want the square root by itself on one side of the equation. Imagine you are peeling layers of an onion to reveal the core – here the square root is your core.

To isolate the square root, carefully shift all other terms in the equation away from it by performing inverse operations. In our example equation, \( \sqrt{4x-1} - 3 = 2 \), we added 3 to both sides to eliminate the -3 next to the square root. So, the equation becomes:

\[\sqrt{4x-1} = 5.\]
Remember, every step you take should lead to having the square root expression alone. This simplification makes it easier to tackle the next steps in solving your equation.

Squaring both sides of an equation

Once you've isolated the square root, the key to eliminating it is to square both sides of the equation. This process helps in removing the square root sign and simplifying the equation to a basic form, such as a linear or quadratic form.

Starting from our isolated equation, \( \sqrt{4x-1} = 5 \), apply the squaring operation as follows:

• Square the square root side: \( (\sqrt{4x-1})^2 = 4x-1 \).
• Square the number on the right: \( 5^2 = 25 \).

Squaring both sides transforms the equation to:

\[4x - 1 = 25.\]
Be careful with this step; once squaring is complete, double-check that all terms are correctly expanded and simplified. This step bridges your path to finally identifying the value of your unknown variable.

Checking solutions

After solving for the variable, always verify your solution by substituting it back into the original equation. This step ensures accuracy and validity since squaring can sometimes introduce extraneous solutions.

In our example, we found \( x = 6.5 \). To check, substitute \( x = 6.5 \) back into the original equation:

\[\sqrt{4(6.5) - 1} - 3 = 2.\]
Calculate the left side step by step:

• Multiply: \( 4 \times 6.5 = 26 \).
• Subtract: \( 26 - 1 = 25 \).
• Find the square root: \( \sqrt{25} = 5 \).
• Subtract 3: \( 5 - 3 = 2 \).

The left side now equals the right side, confirming our solution is correct. Always perform this crucial step to avoid errors and ensure you have the true solution.