Problem 22 - Algebra for College Students

Question

Solve each equation. Don't forget to check each of your potential solutions. \(\sqrt{6 x+5}=\sqrt{2 x+10}\)

Step-by-Step Solution

Answer

The solution is \(x = \frac{5}{4}\).

Step 1: Eliminate the Square Roots

Since the equation involves square roots, begin by eliminating the square roots by squaring both sides of the equation. This yields:\[(\sqrt{6x + 5})^2 = (\sqrt{2x + 10})^2\]which simplifies to:\[6x + 5 = 2x + 10\]

Step 2: Solve for x

To isolate \(x\), first move all terms involving \(x\) to one side of the equation and constant terms to the other side:\[6x - 2x = 10 - 5\]This simplifies to:\[4x = 5\]Divide both sides by 4 to solve for \(x\):\[x = \frac{5}{4}\]

Step 3: Check the Solution

Substitute \(x = \frac{5}{4}\) back into the original square root equation to verify the solution:\[\sqrt{6 \times \frac{5}{4} + 5} = \sqrt{2 \times \frac{5}{4} + 10}\]Calculate each side:\[\sqrt{\frac{30}{4} + 5} = \sqrt{\frac{10}{4} + 10}\]Simplify the left side to:\[\sqrt{\frac{30}{4} + \frac{20}{4}} = \sqrt{\frac{50}{4}}\]and the right side to:\[\sqrt{\frac{10}{4} + \frac{40}{4}} = \sqrt{\frac{50}{4}}\]Both sides simplify to \(\sqrt{\frac{50}{4}}\), so the solution is valid.

Key Concepts

Checking Solutions

Whenever you solve an equation involving square roots, it's crucial to test the potential solutions in the original equation. Checking solutions helps ensure that no extraneous solutions slip through. Extraneous solutions often arise during the process of squaring both sides of an equation, which can potentially introduce non-valid solutions. To check the solution, simply substitute the value of the variable back into the original equation. For instance, in the exercise, the potential solution was calculated as \(x = \frac{5}{4}\). After substituting this value back into the original square root equation \(\sqrt{6x+5} = \sqrt{2x+10}\), both sides of the equation must produce identical results, proving the solution is correct. If they do not match, the solution is not valid, and re-evaluation is needed.

Eliminating Square Roots

Square root equations often appear complex because of the square roots on each side, but we can simplify them by eliminating these roots. The most straightforward method to eliminate square roots is by squaring both sides of the equation. This process will remove the square roots and leave us with a simpler linear or polynomial equation to handle. In our exercise, by squaring both sides of \(\sqrt{6x + 5} = \sqrt{2x + 10}\), we obtained the equation \(6x + 5 = 2x + 10\). Now, it transforms into a much easier problem of solving a linear equation without any square roots involved. This step sets the groundwork for isolating the variable.

Isolating Variable

Isolating the variable is an essential step in solving equations. After eliminating square roots, the equation \(6x + 5 = 2x + 10\) still contains the variable \(x\), which needs to be isolated to find its value. This involves arranging all the terms involving \(x\) on one side of the equation and the constant terms on the other. In the step-by-step solution, the equation was simplified to \(4x = 5\) by subtracting \(2x\) from \(6x\) and \(5\) from \(10\). Finally, we divide each side by 4, finding \(x = \frac{5}{4}\). This straightforward process of isolating the variable helps identify the solution clearly and concisely. Remember, maintaining balance by performing equal operations on both sides of the equation is key.