Problem 20 - Algebra for College Students

Question

Solve each equation. Don't forget to check each of your potential solutions. \(\sqrt{4 x+2}=\sqrt{3 x+4}\)

Step-by-Step Solution

Answer

x = 2.

Step 1: Square Both Sides

To eliminate the square roots in the equation, we need to square both sides of the equation. Given \(\sqrt{4x + 2} = \sqrt{3x + 4}\), when we square both sides, we get:\[(\sqrt{4x + 2})^2 = (\sqrt{3x + 4})^2\]This simplifies to:\[4x + 2 = 3x + 4\]

Step 2: Solve for x

Now that the equation is simplified to \(4x + 2 = 3x + 4\), we need to isolate \(x\). We do this by subtracting \(3x\) from both sides:\[4x + 2 - 3x = 3x + 4 - 3x\]This simplifies to:\[x + 2 = 4\]Next, subtract 2 from both sides to solve for \(x\):\[x + 2 - 2 = 4 - 2\]Thus, \[x = 2\].

Step 3: Check the Solution

To ensure our solution is correct, we substitute \(x = 2\) back into the original equation \(\sqrt{4x + 2} = \sqrt{3x + 4}\):Substitute \(x = 2\):\[\sqrt{4(2) + 2} = \sqrt{3(2) + 4}\]Simplify both sides:\[\sqrt{8 + 2} = \sqrt{6 + 4}\]\[\sqrt{10} = \sqrt{10}\]Since both sides are equal, \(x = 2\) is the correct solution.

Key Concepts

Square Roots

Square roots are an essential concept in mathematics which allow you to determine the original number which, when multiplied by itself, yields another number. In this exercise, we encounter square roots in the equation \(\sqrt{4x + 2} = \sqrt{3x + 4}\). An important property of square roots is that they can be eliminated by squaring both sides of an equation. This is a powerful technique often used in solving equations with square roots, as it transforms root equations into linear or polynomial equations, which are typically simpler to solve.

• Remember that squaring an equation is a key step when dealing with square roots.
• Make sure to apply the square to both sides to keep the equation balanced.

This step helps us transition from an equation with square roots to one without, making it much easier to isolate the variable you are solving for.

Algebraic Equations

To solve algebraic equations like \(4x + 2 = 3x + 4\), we use simple algebraic manipulations. Algebraic equations consist of variables, numbers, and operations. Solving them involves isolating the variable to find its value, which means you might need to perform several operations:

• Subtraction: You subtract terms on both sides to gather like terms.
• Division or multiplication: These operations can be used if needed to solve for the variable.

In our problem, we isolate \(x\) by first subtracting \(3x\) from both sides, yielding \(x + 2 = 4\). Next, subtracting 2 from both sides gives \(x = 2\). Breaking down these operations step-by-step is crucial for understanding how equations transform throughout the solving process.

Checking Solutions

Checking your solutions is a crucial final step in any math problem, especially for equations involving square roots or higher powers. It confirms that your solution satisfies the original equation. After finding \(x = 2\) from our process, we substitute it back into the equation \(\sqrt{4x + 2} = \sqrt{3x + 4}\) to check if it holds true.

• Substitution: Replace \(x\) with the found value in the original equation to verify both sides match.
• Comparison: Check that the left-hand side equals the right-hand side to confirm correctness.

For our solution, substituting \(x = 2\) results in \(\sqrt{10} = \sqrt{10}\), confirming the correctness of \(x = 2\). This step ensures that potential errors made in the calculation process are caught and corrected before finalizing your answer.