Problem 39 - Algebra 2 and Trigonometry

Question

In \(39-42,\) solve and check each equation. $$ 5 x-\sqrt{3}=\sqrt{48} $$

Step-by-Step Solution

Answer

The solution is \(x = \sqrt{3}\).

Step 1: Simplify the Equation

First, simplify the given equation by calculating the square root of 48. Notice that 48 can be written as a product of its prime factors: 48 = 16 * 3. This simplifies the square root: \( \sqrt{48} = \sqrt{16 \cdot 3} = \sqrt{16} \cdot \sqrt{3} = 4\sqrt{3} \). Substitute this back into the equation to get:\[ 5x - \sqrt{3} = 4\sqrt{3} \]

Step 2: Isolate the Variable

To isolate \(x\), we need to move \(\sqrt{3}\) to the other side of the equation. Add \(\sqrt{3}\) to both sides:\[ 5x = 4\sqrt{3} + \sqrt{3} \] Combine like terms on the right side:\[ 5x = 5\sqrt{3} \]

Step 3: Solve for x

Now solve for \(x\) by dividing both sides of the equation by 5:\[ x = \frac{5\sqrt{3}}{5} \] This simplifies to:\[ x = \sqrt{3} \]

Step 4: Check the Solution

To verify the solution, replace \(x\) with \(\sqrt{3}\) in the original equation:\[ 5(\sqrt{3}) - \sqrt{3} = \sqrt{48} \]This simplifies as:\[ 5\sqrt{3} - \sqrt{3} = 4\sqrt{3} \]Simplifying further:\[ 4\sqrt{3} = 4\sqrt{3} \]The equation holds true, confirming that \(x = \sqrt{3}\) is the correct solution.

Key Concepts

Simplifying Square Roots

Simplifying square roots is a key math concept that enables you to make complex equations easier to manage. The goal is to express a square root in its simplest form. When dealing with numbers like 48, you start by identifying its prime factors.

• First, write 48 as a product of its prime factors: 48 = 16 * 3.
• Notice that 16 is a perfect square (\( \sqrt{16} = 4 \)), allowing further simplification when it becomes a factor under the square root.
• This means you can break down \( \sqrt{48} \) into \( \sqrt{16} \times \sqrt{3} \).
• By simplifying, it becomes \( 4\sqrt{3} \).

Understanding this process is crucial because it helps you to simplify complicated numbers easily and is often used in solving diverse math problems.

Isolating Variables

Isolating variables in an equation is like finding the key piece of a puzzle, and it's an important step to solve for unknown values. Once we simplify the square roots, we look to isolate the variable, often represented by \( x \).

• In the equation \( 5x - \sqrt{3} = 4\sqrt{3} \), our task is to get \( x \) by itself on one side.
• To do this, we add \( \sqrt{3} \) to both sides, balancing the equation and maintaining equality.
• This transforms the equation to \( 5x = 5\sqrt{3} \), leaving \( x \) ready to be solved.
• By dividing both sides by 5, you isolate \( x \), finding \( x = \sqrt{3} \).

Isolating variables can simplify the solution and it's a fundamental technique used across algebraic problems.

Equation Verification

Verification is the final step of solving square root equations. After solving for the variable, it's essential to check that your derived solution holds true in the original equation. This confirms the correctness and accuracy of your solution.

• Take the derived solution, \( x = \sqrt{3} \), and substitute it back into the original equation \( 5x - \sqrt{3} = \sqrt{48} \).
• By substituting, we get \( 5(\sqrt{3}) - \sqrt{3} = \sqrt{48} \).
• Simplify further: \( 5\sqrt{3} - \sqrt{3} = 4\sqrt{3} \).
• Finally, compare both sides: \( 4\sqrt{3} = 4\sqrt{3} \), confirming that both sides are equal.

Once the original and substituted equations match, you can be confident that \( x = \sqrt{3} \) is indeed the correct solution. This step is very crucial as it validates your entire solving process.