Problem 50 - Intermediate Algebra

Question

Solve each equation. See Example 7. $$ \sqrt{6-2 x}=4 \sqrt{x-3} $$

Step-by-Step Solution

Answer

The solution to the equation is \(x = 3\), which checks out as valid.

Step 1: Eliminate the Square Roots

To eliminate the square roots, square both sides of the equation. This gives us:\[ ( \sqrt{6 - 2x} )^2 = ( 4 \sqrt{x - 3} )^2 \]Simplifying both sides, we get:\[ 6 - 2x = 16(x - 3) \]

Step 2: Expand the Right Side

Expand the right side of the equation:\[ 6 - 2x = 16x - 48 \]

Step 3: Combine Like Terms

Add \(2x\) to both sides to get variables on one side and add \(48\) to both sides to isolate the constant terms on the other side:\[ 6 + 48 = 16x + 2x \]Simplify:\[ 54 = 18x \]

Step 4: Solve for x

Divide both sides by \(18\) to solve for \(x\):\[ x = \frac{54}{18} \]Simplify:\[ x = 3 \]

Step 5: Verify the Solution

Substitute \(x = 3\) back into the original equation to verify that the solution is valid:\[ \sqrt{6 - 2(3)} = 4 \sqrt{3 - 3} \]The equation simplifies to:\[ \sqrt{0} = 4 \sqrt{0} \]Both sides equal zero, so the solution is verified as correct.

Key Concepts

Solving Radical Equations

When faced with a radical equation, the main goal is to remove the square roots so we can solve the rest of the equation more easily. Radical equations are equations where the variable is found inside a square root or any other root. Here's the basic approach to solving them:

• First, isolate the radical on one side of the equation if there are multiple terms.
• Next, eliminate the square root by squaring both sides of the equation. Be careful: this step can introduce extraneous solutions!
• After eliminating the square root, the equation usually simplifies into a linear or quadratic one. You proceed to solve it just like any other equation.

Remember, the goal is to simplify the equation until you have isolated the variable. Once done, don’t forget to check your solution by substituting it back into the original equation. Failing to check could mean accepting a mathematically incorrect solution.

Square Roots

The square root is a fundamental mathematical concept, which asks "what number, when multiplied by itself, gives this?" In radical equations, spotting the square root term is crucial. For example, in the equation \( \sqrt{6 - 2x} = 4 \sqrt{x - 3} \), both sides contain square root expressions.

Working with square roots requires utilizing some important properties:

• Simplifying the square root of perfect squares: For instance, \( \sqrt{16} = 4 \)
• Square roots can be manipulated with multiplication and division: \( \sqrt{a} \times \sqrt{b} = \sqrt{a \times b} \)
• Always remember: squaring a square root nullifies the square root operation: \( \sqrt{a}^2 = a \)

Understanding and maneuvering these properties is essential when you work to solve equations or simplify expressions containing square roots.

Equation Verification

Verifying an equation is a vital step in solving any algebraic equation. This ensures that your solution holds true and is not an artifact of the solving process or any operations applied. Here's how to verify your solutions:

• Take the solution you found and substitute it back into the original equation.
• Simplify both sides of the equation to see if they match.
• If they match, then congratulations! Your solution is correct. If they don't, you might have an extraneous solution or may have made a calculation mistake along the way.

In our example, substituting \( x = 3 \) back into \( \sqrt{6 - 2x} = 4 \sqrt{x - 3} \) led both sides to simplify to 0, confirming the solution. Verification is crucial for confirming the validity of your results and reinforcing your mathematical process.