Problem 21 - Algebra 1

Question

Solve the equation. Check for extraneous solutions. $$\sqrt{6 x}-13=23$$

Step-by-Step Solution

Answer

The solution to the equation \( \sqrt{6x} - 13 = 23 \) is \( x = 216 \)

Step 1: Isolate the square root

Add 13 to both sides of the equation to isolate the square root. This will give:\( \sqrt{6x} = 23 + 13 \)\( \sqrt{6x} = 36 \)

Step 2: Square both sides

To get rid of the square root, square both sides of the equation:\((\sqrt{6x})^2 = 36^2 \)This simplifies to:\( 6x = 1296 \)

Step 3: Solve for x

Divide both sides by 6 to isolate x:\( x = 1296 / 6 \)\( x = 216 \)

Step 4: Check Solution

Substitute x = 216 into the original equation to check if it's a valid solution:\( \sqrt{6*216} - 13 ?= 23 \)The left side simplifies to:\( \sqrt{1296} - 13 = 36 - 13 = 23 \)Since the left side equals the right side, x = 216 is a valid solution to the given equation.

Key Concepts

Understanding Square Roots in Equations

When solving equations that involve square roots, it's essential to isolate the square root term. The square root symbol \( \sqrt{} \) represents a value that, when multiplied by itself, gives the original number inside the square root. For instance, in the equation \( \sqrt{6x} - 13 = 23 \), we start by isolating \( \sqrt{6x} \) on one side: \( \sqrt{6x} = 36 \). This ensures clarity in the subsequent steps.
A critical technique in such scenarios is squaring both sides of the equation. Squaring effectively removes the square root, transforming \( \sqrt{6x} = 36 \) into \( 6x = 1296 \). This conversion simplifies the process of solving the equation, making it more straightforward.
Remember:

• Always isolate the square root first to avoid mistakes.
• Squaring both sides removes the square root but requires attention to maintain equation balance.

Mastering this process is fundamental to handling more complex equations involving square roots.

Dealing With Extraneous Solutions

Extraneous solutions are results that emerge from the solving process but do not satisfy the original equation. These solutions often appear when both sides of an equation are squared, as squaring can introduce values that don't hold true in the context of the initial problem.
In our example, squaring the equation \( \sqrt{6x} = 36 \) to obtain \( 6x = 1296 \) leads us to calculate \( x = 216 \). We must beware that squaring could introduce extraneous solutions, which means the solution might not satisfy the original equation.
Why worry about extraneous solutions?

• Squaring both sides can introduce solutions that are not valid.
• Verifying with the original equation ensures the solution is correct.

Always take the extra step to check, as solving without verification can lead to accepting incorrect solutions.

Checking Your Solution

Checking your solution is a crucial step in solving equations, especially when square roots and squaring are involved. This process helps confirm whether the solution satisfies the original equation or if it is an extraneous result.
For example, let's verify the solution \( x = 216 \) by substituting it back into the original equation \( \sqrt{6 \times 216} - 13 = 23 \). Simplifying the left side, we get \( \sqrt{1296} - 13 = 36 - 13 = 23 \), which matches the right side of the equation.
Tips for a successful check:

• Re-insert the solved value into the original equation.
• Simplify to ensure both sides of the equation are equal.
• If they are not, re-evaluate for possible mistakes or extraneous solutions.

This step guarantees not only the correctness of your solution but also strengthens your problem-solving skills, ensuring precision and accuracy in mathematics.