Problem 24 - Algebra 1

Question

Solve the equation. Check for extraneous solutions. $$\sqrt{5 x+1}+2=6$$

Step-by-Step Solution

Answer

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

Step 1: Isolate the square root

Subtract 2 from both sides of the equation to isolate the square root, giving \(\sqrt{5x + 1} = 6 - 2 =4\).

Step 2: Square both sides

Square both sides of the equation to remove the square root: \((\sqrt{5x + 1})^2 = 4^2\), simplifying to \(5x + 1 = 16\).

Step 3: Solve for x

Solve for x in the equation from the previous step by subtracting 1 from both sides and then dividing by 5: \(5x = 16 - 1 = 15\), so \(x = 15/5 = 3\).

Step 4: Check for extraneous solutions

Substitute \(x = 3\) into the original equation to check if it's a valid solution: \(\sqrt{5*3 + 1} + 2 = \sqrt{16} + 2 = 4 + 2 = 6\). Since both sides of the equation are equal, x = 3 is a valid solution.

Key Concepts

Extraneous Solutions

When solving equations involving square roots, it's crucial to watch out for extraneous solutions. These are solutions that emerge from the process of solving the equation but don't actually satisfy the original equation.

Why do they appear? It usually happens after squaring both sides of an equation, as this operation can introduce solutions that weren't there before. This squaring step ignores the fact that the square root function originally required the quantity under the root to be non-negative. Therefore, after finding potential solutions, you must always substitute them back into the original equation to verify their validity.

If the original equation holds true after the substitution, then the solutions are genuine. If not, they are considered extraneous and should be discarded.

Isolating the Variable

To effectively solve square root equations, the initial step is to isolate the variable. This means you want to have the square root expression by itself on one side of the equation.

Here's how to do it: First, simplify the equation as much as possible by doing basic algebraic operations such as adding or subtracting terms on both sides. Once you have the square root on one side and all other terms on the other, you are set to proceed to the next steps of finding your solution.

Remember, this initial maneuver sets the stage for eliminating the square root, and ensuring precision here prevents complications in subsequent steps.

Square Both Sides of an Equation

Once you have isolated the square root on one side of the equation, the next move is to square both sides of the equation. This step is used to get rid of the square root symbol and make the variable easier to solve for.

Squaring is simply multiplying a number by itself, and when you square a square root, you're left with just the value inside the root. But beware, squaring can affect the balance of an equation as it introduces the possibility of those pesky extraneous solutions we mentioned earlier.

After squaring both sides, you'll continue to solve for the variable using standard algebra - just don't forget to check for those extraneous solutions once you've found your answers!