Problem 44 - Algebra 2 and Trigonometry

Question

Solve and check each equation. \(5 x-\sqrt{12}=\sqrt{108}-3 x\)

Step-by-Step Solution

Answer

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

Step 1: Simplify the Square Roots

First, simplify the square roots in the equation. The square root of 12 can be simplified as \( \sqrt{12} = \sqrt{4 \times 3} = 2\sqrt{3} \). Similarly, the square root of 108 can be simplified as \( \sqrt{108} = \sqrt{36 \times 3} = 6\sqrt{3} \). Substitute these back into the equation to get: \( 5x - 2\sqrt{3} = 6\sqrt{3} - 3x \).

Step 2: Move Variable Terms to One Side

Next, move all the variable terms to one side of the equation and constant terms to the other side. Add \( 3x \) to both sides: \( 5x + 3x - 2\sqrt{3} = 6\sqrt{3} \). This simplifies to \( 8x - 2\sqrt{3} = 6\sqrt{3} \).

Step 3: Isolate the Variable Term

To isolate the variable term, add \( 2\sqrt{3} \) to both sides of the equation: \( 8x = 6\sqrt{3} + 2\sqrt{3} \). Simplifying the right side gives: \( 8x = 8\sqrt{3} \).

Step 4: Solve for the Variable

Divide both sides of the equation by 8 to solve for \( x \): \( x = \frac{8\sqrt{3}}{8} \). Simplifying this gives \( x = \sqrt{3} \).

Step 5: Verify the Solution

Substitute \( x = \sqrt{3} \) back into the original equation to verify the solution: \[ 5 (\sqrt{3}) - \sqrt{12} = \sqrt{108} - 3(\sqrt{3}) \]. Simplifying each side, we have: \( 5\sqrt{3} - 2\sqrt{3} = 6\sqrt{3} - 3\sqrt{3} \). Simplified, both sides equal \( 3\sqrt{3} \), thus confirming our solution is correct.

Key Concepts

Simplify Square Roots

Simplifying square roots is an essential technique when solving equations. It involves breaking down the radicand (the number inside the square root) into its prime factors and simplifying. For example, consider the expression \( \sqrt{12} \). Notice that 12 can be expressed as the product of 4 and 3, where 4 is a perfect square.

By using the property \( \sqrt{a \times b} = \sqrt{a} \times \sqrt{b} \), we can rewrite:

• \( \sqrt{12} = \sqrt{4 \times 3} \)
• This simplifies further to \( \sqrt{4} \times \sqrt{3} = 2\sqrt{3} \).

Similarly, simplifying \( \sqrt{108} \) involves breaking down 108 as \( 36 \times 3 \), where 36 is also a perfect square, giving us \( 6\sqrt{3} \).
Simplification is crucial because it makes the equation easier to handle by reducing it to terms involving rational numbers and smaller radicals.

Isolating Variables

Isolating the variable is a key operation in solving equations. It involves moving all instances of a variable to one side of the equation and the constants to the other. Here, let's see how this works:

Given the equation from the exercise: \[ 5x - 2\sqrt{3} = 6\sqrt{3} - 3x \]We aim to gather all terms with \( x \) on the left. By adding \( 3x \) to both sides, we merge the variable terms:

• \( 5x + 3x - 2\sqrt{3} = 6\sqrt{3} \)
• This results in \( 8x - 2\sqrt{3} = 6\sqrt{3} \).

Moving terms around in this manner allows you to focus on solving for \( x \), by reducing clutter and concentrating similar terms together.

Verifying Solutions

Verification is the process where you check whether the solution obtained is correct by substituting it back into the original equation. This is crucial because it confirms that no mistakes were made during the manipulations. In this instance, we check the solution \( x = \sqrt{3} \):

• Substitute into the original equation: \( 5(\sqrt{3}) - \sqrt{12} = \sqrt{108} - 3(\sqrt{3}) \)
• Simplify both sides:
• Left side becomes: \( 5\sqrt{3} - 2\sqrt{3} = 3\sqrt{3} \)
• Right side equals: \( 6\sqrt{3} - 3\sqrt{3} = 3\sqrt{3} \)

Both sides are equal, thus verifying that our solution is indeed accurate. Always perform this check to ensure reliability.

Algebraic Manipulation

Algebraic manipulation involves rearranging and combining terms following the rules of algebra to achieve a solution. It requires understanding operations like addition, subtraction, and division applied systematically to each side of the equation. Consider how manipulation occurs step-by-step:

• Start with: \( 8x - 2\sqrt{3} = 6\sqrt{3} \)
• Add \( 2\sqrt{3} \) to both sides to focus on \( x \):
  \( 8x = 6\sqrt{3} + 2\sqrt{3} \).
• Combine like terms results in \( 8x = 8\sqrt{3} \).
• Finally, divide by 8 to solve for \( x \):
  \( x = \frac{8\sqrt{3}}{8} = \sqrt{3} \)

Employing these techniques helps streamline the process and sidestep common arithmetic inaccuracies, enhancing both efficiency and clarity.