Problem 30 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 1-50: Solve the equation. $$ 2 \sqrt{x}-\sqrt{x-3}=\sqrt{5+x} $$

Step-by-Step Solution

Answer

The solution to the equation is \(x=4\).

Step 1: Isolate one square root term

To start solving the equation \(2 \sqrt{x} - \sqrt{x-3} = \sqrt{5+x}\), we will move \(\sqrt{x-3}\) to the right side of the equation. This gives us: \(2 \sqrt{x} = \sqrt{5+x} + \sqrt{x-3}\).

Step 2: Square both sides of the equation

Next, eliminate the square root by squaring both sides of the equation. The equation \(2 \sqrt{x} = \sqrt{5+x} + \sqrt{x-3}\) becomes \((2 \sqrt{x})^2 = (\sqrt{5+x} + \sqrt{x-3})^2\), which simplifies to \(4x = (5+x) + 2 \sqrt{(5+x)(x-3)} + (x-3)\).

Step 3: Simplify the equation

Simplify the resulting equation from the previous step: \(4x = 5 + x + 2 \sqrt{(5+x)(x-3)} + x - 3\). This simplifies further to \(2x = 2 + 2 \sqrt{(5+x)(x-3)}\).

Step 4: Square both sides again

To eliminate the remaining square root, we isolate it and square both sides again: first divide by 2, giving \(x = 1 + \sqrt{(5+x)(x-3)}\), and then square both sides to get \(x^2 = 1^2 + 2\cdot1\cdot\sqrt{(5+x)(x-3)} + (5+x)(x-3)\).

Step 5: Rearrange and solve for x

Rearrange the equation obtained from squaring: \(x^2 - (5x - 15 - x^2) = 2 \sqrt{(5+x)(x-3)}\). Simplify to \(0 = 10 - 3x + 2 \sqrt{(5+x)(x-3)}\). This simplifies to finding the right value of \(x\).

Step 6: Evaluate likely solutions

Assume integers for \(x\) and check solutions. Try \(x = 4\). Substituting into the equation, \(2\sqrt{4} - \sqrt{4-3} = \sqrt{5+4}\) checks out: \(4 - 1 = 3\), the same as \(\sqrt{9} = 3\).

Step 7: Verify solutions

Always verify any potential solution by substituting it back into the original equation. For \(x = 4\), it satisfies the equation \(2 \sqrt{4} - \sqrt{4-3} = \sqrt{5+4}\), confirming that \(x = 4\) is the correct solution.

Key Concepts

Squaring Equations

One of the most essential tools in solving radical equations is squaring, which helps eliminate the square roots to simplify the equation. In the provided exercise, squaring was used twice to help simplify the problem.
When squaring an equation, you take both sides of the equation and square them. This is essential when you have terms with square roots, as squaring cancels out those roots. For example, in the equation \(2 \sqrt{x} = \sqrt{5+x} + \sqrt{x-3}\), squaring transforms it into a polynomial-like form:

• (2 \( \sqrt{x})^2 = (\sqrt{5+x} + \sqrt{x-3})^2\) becomes \(4x = (5+x) + 2 \sqrt{(5+x)(x-3)} + (x-3)\).

This technique is powerful, but it is important to remember that sometimes it can introduce extraneous solutions—solutions that work in the squared equation, but not in the original. Therefore, further verification is required after finding solutions.

Step-by-Step Algebra

Working through complex problems step by step is crucial to simplifying and solving them correctly. In the given solution, each step systematically tackles parts of the equation and reduces complexity.
The initial steps revolve around isolating square root terms. For instance, the step \(2 \sqrt{x} - \sqrt{x-3} = \sqrt{5+x}\) was simplified to \(2 \sqrt{x} = \sqrt{5+x} + \sqrt{x-3}\) by moving terms. This clear strategy of isolating and simplifying piece by piece helps avoid confusion and errors.

• Once terms are isolated, the next steps involve squaring the equation to remove square roots, making the problem more straightforward.
• Algebraic rearrangements and simplifications form the core part of understanding and solving these types of problems.

By logically processing through each step, students build a strong methodology to tackle other similar problems, gaining confidence in their algebraic skills.

Verification of Solutions

After obtaining prospective solutions, verification is a vital step in solving equations effectively and accurately. Verification involves substituting your solution back into the original equation to ensure it satisfies all parts of the equation.
For instance, in this exercise, after determining that \(x = 4\) could be a valid solution, the final step of the process checked this value by substituting it back into the original equation.

• Substituting \(x = 4\) gives \(2 \sqrt{4} - \sqrt{4-3} = \sqrt{5+4}\).
• Calculating gives \(4 - 1 = 3\) which matches \(\sqrt{9} = 3\).

The successful verification confirms that \(x = 4\) is a legitimate solution, serving as an affirmation of the problem-solving process. This step catches any incorrect solutions that might not fit into the original context of the problem due to issues like squaring errors.

Simplifying Expressions

Simplifying complex expressions into more manageable forms is an indispensable part of algebra. For the exercise, it involves breaking down complex partial expressions for easier solving.
After the initial squaring, example simplification is seen when reducing \(4x = 5 + x + 2 \sqrt{(5+x)(x-3)} + x - 3\) to \(2x = 2 + 2 \sqrt{(5+x)(x-3)}\). Eliminating redundant terms and simplifying coefficients transforms the equation drastically.

• Further simplification then isolates the variable, making a complex-looking problem approachable and solvable.
• This process involves combining like terms and factoring where possible, usually iteratively going through the equation from the most complex to simplest parts.

In mastering simplification, students train themselves to see patterns and symmetries in algebra, allowing more complex problems to be handled with ease.