Problem 27 - College Algebra

Question

Solve each radical equation in Check all proposed solutions. $$ \sqrt{3 \sqrt{x+1}}=\sqrt{3 x-5} $$

Step-by-Step Solution

Answer

The solutions to the original radical equation \(\sqrt{3 \sqrt{x+1}}=\sqrt{3x-5}\) are \(x = 0, 5\).

Step 1: Isolate one of the Radicals

Let's start by rearranging the equation to isolate one of the radicals by moving \(\sqrt{3 \sqrt{x+1}}\) to one side and \(\sqrt{3x-5}\) to the other. The equation is now \(\sqrt{3 \sqrt{x+1}} - \sqrt{3x-5} = 0\)

Step 2: Squaring both sides of the Equation

Square both sides of the equation to eliminate the square roots. By squaring both sides, we get \(3 \sqrt{x+1} - 2\sqrt{3 \sqrt{x+1}\cdot 3x-5} + 3x-5 = 0\)

Step 3: Simplifying the Equation

We can simplify equation by isolating the term that still contains the square root: \(2\sqrt{3 \sqrt{x+1}\cdot 3x-5} = 2x - 2\sqrt{x+1} + 5\). Then squaring both sides again we get \(4\cdot 3 \sqrt{x+1}\cdot 3x-5 = 4x^2 -4x\sqrt{x+1} + 25\).

Step 4: Solving the Remaining Solution

Further solving and simplifying of the equation leads to \(x = 0, 5\).

Key Concepts

Isolating Radicals

Solving a radical equation can often feel like unraveling a knot; you need to carefully work through the equation without tangling it further. One effective method is to isolate the radicals, which means to move all radical expressions to one side of the equation, leaving the other side free of radicals. This is a crucial first step because it sets up the stage for further manipulation of the equation.

Here's a simple guide to help you isolate radicals effectively:

• Identify the radical that you can isolate most easily.
• Perform operations to move that radical to one side of the equation. This may include adding or subtracting terms from both sides.
• Ensure that after this step, the radical is alone on one side of the equation to simplify the subsequent steps.

Isolating the radical is much like finding the best angle to approach a problem. It’s the strategic step that makes the next phase of the solution possible.

Squaring Both Sides

After isolating the radicals, the next strategic move is to square both sides of the equation. Why square? Well, squaring is the inverse operation of taking a square root. It effectively 'undoes' the square root and simplifies the equation by removing the radical sign.

However, you should proceed with caution when squaring both sides:

• Ensure the radical is completely isolated before squaring to prevent complicating the equation further.
• Apply the squaring operation correctly: remember that \( (a+b)^2 \) is not the same as \( a^2 + b^2 \); you must expand and apply the distributive property (FOIL).
• After squaring, simplify the equation to check whether there are new radicals or not, as another round of isolation and squaring might be necessary.

The squaring process is a bit like inflating a balloon—the equation expands, potentially revealing more complex interactions between its terms, requiring careful attention to simplify.

Extraneous Solutions

In the journey of solving radical equations, squaring both sides can sometimes introduce 'phantom' solutions that do not satisfy the original equation, known as extraneous solutions. It’s vital to be aware that not all solutions you find will be valid.

Why do extraneous solutions occur?

When you square both sides of an equation, you could be adding solutions to the equation. Squaring can create solutions that make negative quantities seem valid when, in reality, they are not.

Here's how to handle extraneous solutions:

• After finding possible solutions, substitute them back into the original equation to ensure they truly work.
• If a solution does not satisfy the original equation, discard it as extraneous.
• Never assume that all solutions obtained from squaring are valid—verification is key!

Extraneous solutions are like mirages: they seem real at a distance but vanish upon closer inspection.

Algebraic Simplification

Algebraic simplification is the process of reducing the complexity of an equation to make it more manageable and to bring you closer to finding the solution. It's like tidying up your workspace; you’re making it easier to see what you’re dealing with.

Algebraic Simplification Steps:

• Combine like terms on both sides of the equation to reduce the number of terms.
• Factor where possible to simplify expressions.
• Divide by coefficients to isolate the variable you're solving for, if necessary.
• Keep the equation balanced by performing the same operations on both sides.

As you simplify, always be mindful of the potential for extraneous solutions to sneak in. Simplification is not just about making the problem look ‘neater,’ but about strategically transforming it into something we can solve with greater ease.