Problem 116 - College Algebra

Question

In solving \(\sqrt{3 x+4}-\sqrt{2 x+4}-2,\) why is it a good idea to isolate a radical term? What if we don't do this and simply square each side? Describe what happens.

Step-by-Step Solution

Answer

By isolating a radical term, squaring both sides results in a simpler equation that can be easier to solve. If we don't do this, the equation becomes more complex and harder to solve due to the presence of a square root product after squaring.

Step 1: Isolate a radical

Write the given equation and isolate a radical term by bringing the other terms to the other side of the equation. This will take the form: \(\sqrt{3x+4} = \sqrt{2x + 4} + 2\)

Step 2: Squaring both sides

Now that we have a radical isolated, we square both sides of the equation to eliminate the square root. This results in \(3x + 4 = 2x + 4 + 4\sqrt{2x+4} + 4\) which simplifies further to \(x = 4\sqrt{2x+4}\). Continued simplification and squaring of both sides led to simpler and solvable equations.

Step 3: Squaring without isolation

Now, let's investigate what happens if we square without isolating a term. Squaring the original equation \(\sqrt{3x+4} - \sqrt{2x+4} = 2\) leads to a very complex equation, \(3x + 4 - 2\sqrt{(3x+4)(2x+4)} + 2x + 4 = 4\), which is not ideal for solving algebraic equations due to the square root product obtained from cross multiplication.

Key Concepts

Isolate Radical

When it comes to solving radical equations, isolating the radical is a vital step. A radical equation includes a variable inside a square root. For example, in the equation \(\sqrt{3x+4} = \sqrt{2x + 4} + 2\), the radical term \(\sqrt{3x+4}\) needs to be isolated on one side of the equation. By isolating the radical, you simplify the equation, making it easier to manipulate.
Isolating a radical involves:

• Rearranging the equation to have a single radical term on one side
• Moving any other terms to the opposite side

This helps to reduce complexity and prepares the equation for further operations, like squaring both sides. It's like clearing the playing field before you start solving deeper parts of the equation. Without isolating the radical first, subsequent steps become cumbersome and can lead to errors.

Squaring Both Sides

Once a radical is isolated, the next step is often to square both sides of the equation. This helps to remove the square root, leaving you with a more straightforward algebraic expression.
Here's how it works with our example: from the isolated form, \(\sqrt{3x+4} = \sqrt{2x + 4} + 2\), squaring both sides results in \(3x + 4 = 2x + 4 + 4\sqrt{2x+4} + 4\).
The process includes:

• Mimicking each side when squaring
• Expanding the expressions that involve distribution

However, always remember:

• Squaring can introduce extraneous solutions or artifacts that aren't solutions to the original equation
• It's crucial to check your work afterward

This step can give you a more manageable equation to simplify further or solve directly.

Algebraic Simplification

Algebraic simplification is the key step after squaring both sides of a radical equation. This process involves simplifying the equation into a form that can be easily solved.
After squaring both sides, in the example \(3x + 4 = 2x + 4 + 4\sqrt{2x+4} + 4\), the goal is to simplify the expression involving \(x\). You'll:

• Combine like terms
• Isolate the variable, if possible
• Solve the resulting algebraic equation using basic algebra steps

Effective simplification makes the equation clear and reveals the solution. Remember to verify the solution by plugging it back into the original equation to ensure it does not arise from any artifacts introduced during squaring. This last step confirms the solution is both valid and applicable to the original radical equation.