Problem 23 - Algebra for College Students

Question

Solve each equation. Don't forget to check each of your potential solutions. \(\sqrt{3 x-2}-\sqrt{x+4}=0\)

Step-by-Step Solution

Answer

The solution is \( x = 3 \).

Step 1: Isolate One Square Root

Start with the given equation \( \sqrt{3x - 2} - \sqrt{x + 4} = 0 \). Add \( \sqrt{x + 4} \) to both sides to isolate one of the square roots:\[ \sqrt{3x - 2} = \sqrt{x + 4} \]

Step 2: Square Both Sides

Square both sides of the equation to remove the square roots:\[ (\sqrt{3x - 2})^2 = (\sqrt{x + 4})^2 \]This simplifies to:\[ 3x - 2 = x + 4 \]

Step 3: Solve for x

Rearrange the equation to solve for \( x \):\[ 3x - x = 4 + 2 \]\[ 2x = 6 \]Divide both sides by 2:\[ x = 3 \]

Step 4: Check the Solution

Substitute \( x = 3 \) back into the original equation to verify it satisfies both square roots:\[ \sqrt{3(3) - 2} - \sqrt{3 + 4} = \sqrt{9 - 2} - \sqrt{7} = \sqrt{7} - \sqrt{7} = 0 \]Since both sides are equal, \( x = 3 \) is indeed a solution.

Key Concepts

Isolating One Square Root

Starting with an equation containing two square root expressions can seem intimidating at first. However, the key strategy to tackle such problems is focusing on isolating one square root to simplify the process. Let's think about our original equation:

• \( \sqrt{3x - 2} - \sqrt{x + 4} = 0 \)

The goal is to have just one square root on one side of the equation. In this case, we can achieve this by adding \( \sqrt{x + 4} \) to both sides. The adjusted equation becomes:

• \( \sqrt{3x - 2} = \sqrt{x + 4} \)

By isolating one square root, the problem becomes much simpler to handle, as you are then setting two expressions equal to each other without the complications of mixed terms. This step is crucial because it allows us to move on to squaring each side of the equation, eliminating the square roots altogether and reducing the complexity of the problem.

Square Both Sides

Once you have successfully isolated a square root on each side of the equation, the next logical step is to square both sides. This technique allows us to eliminate the square roots and transform the equation into a polynomial form. So now our updated equation:

• \( \sqrt{3x - 2} = \sqrt{x + 4} \)

Can be squared to remove the radicals:

• \((\sqrt{3x - 2})^2 = (\sqrt{x + 4})^2\)

Which simplifies the problem to:

• \( 3x - 2 = x + 4 \)

By squaring each side, we're converting the equation from involving roots to one that is easier to handle and solve algebraically. The expression now looks linear, and we can proceed to isolate the variable \( x \) using basic algebraic techniques. Squaring both sides is a crucial step to transition from a complex equation to something much easier to work with.

Checking Solutions

Solving the equation is only part of the process. An essential final step is verifying the solution to ensure it is correct. The importance of this step cannot be overstated as, during the squaring process, extraneous solutions (solutions that don't satisfy the original equation) may be introduced. After finding \( x = 3 \), we substitute it back into the original equation:

• \( \sqrt{3(3) - 2} - \sqrt{3 + 4} = 0 \)

This becomes:

• \( \sqrt{9 - 2} - \sqrt{7} = \sqrt{7} - \sqrt{7} = 0 \)

Both sides simplify to zero, confirming that \( x = 3 \) is indeed a valid solution. Always remember, when dealing with square root equations, it’s critical to check back against the original equation. This way, you can be confident that your solution isn't just a mistake made along the way!