Problem 27 - Algebra 2

Question

Solve. Check for extraneous solutions. \((3 x+2)^{\frac{1}{2}}-(2 x+7)^{\frac{1}{2}}=0\)

Step-by-Step Solution

Answer

The equation has no valid solutions as our calculated solution is extraneous.

Step 1: Isolate one of the square root expressions

To isolate \( (2x+7)^{1/2} \), we can begin by moving it to the right hand side of the equation. Thus, the equation becomes \( (3x+2)^{1/2} = (2x+7)^{1/2} \).

Step 2: Square both sides

Square both sides in order to get rid of the square roots: \( ((3x+2)^{1/2})^2 = ((2x+7)^{1/2})^2 \). This simplifies to \( 3x+2 = 2x+7 \).

Step 3: Rearrange and solve the equation

Firstly, we will rearrange the equation: \(3x - 2x = 7 - 2 \) which simplifies to \( x = 5 \).

Step 4: Check for extraneous solutions

Substitute the value of x = 5 from our solution into the original equation to check for extraneous solutions. We get: \( (3*5+2)^{1/2} = (2*5+7)^{1/2} \) which simplifies to \( 11^{1/2} = 7^{1/2} \), which is not true. Thus, our solution x=5 is an extraneous solution which is not valid for the original equation.

Key Concepts

Understanding Square Root Equations

Square root equations are mathematical expressions that include the square root of a variable or number. Solving these can be tricky, as the presence of square roots requires careful manipulation to find the solution. When dealing with square root equations, the main goal is to eliminate the square root so that you are left with an algebraic equation that you can solve more straightforwardly. In this exercise, we have an equation with two square root terms, \((3x+2)^{1/2}\) and \((2x+7)^{1/2}\). By isolating them and then squaring both sides, we can simplify the equation to make it easier to solve. This process needs to be undertaken with caution as it can often produce extraneous solutions.

Mastering Equation Solving

Equation solving is a critical skill in mathematics, allowing us to find the values of variables that satisfy given conditions. In our exercise, after isolating and squaring the square root terms, we ended up with a simplified linear equation: \(3x+2 = 2x+7\). Solving this type of equation involves basic algebraic manipulation such as adding, subtracting, multiplying, or dividing both sides to isolate the variable in question. For this exercise:

• Subtract \(2x\) from both sides to get \(x\) by itself on one side.
• Simplify the constants on the other side by subtraction.
• The resulting equation \(x = 5\) seemed like a valid solution.

This step solves the equation but doesn’t confirm it satisfies the original conditions, which is why checking solutions is crucial.

Checking Solutions for Validity

In mathematics, solutions must always be validated to ensure they are correct, especially in square root equations, which can have extraneous solutions. Extraneous solutions are results that emerge from the solving process but do not satisfy the original equation. In our exercise, the supposed solution \(x = 5\) failed to meet the original condition of the equation when substituted back. After working through the square roots:

• Substitute \(x = 5\) back into the original square root equation: \((3*5+2)^{1/2} - (2*5+7)^{1/2} = 0\).
• This leads to \(11^{1/2} = 7^{1/2}\), a false statement.
• Thus, \(x = 5\) isn't a valid solution, highlighting the importance of this checking step.

This check ensures that what seems like a solution actually fulfills the equation's requirements.

Isolating Variables Effectively

Isolating variables is a vital part of solving equations. The goal is to get the variable by itself on one side of the equation, making it easier to both understand and solve. In square root equations, this often involves moving terms from one side to another before eliminating the square root. For example:

• Initially, the equation had \((3x+2)^{1/2} - (2x+7)^{1/2} = 0\).
• To progress, we moved \((2x+7)^{1/2}\) to the other side, resulting in \((3x+2)^{1/2} = (2x+7)^{1/2}\).
• This repositioning is critical as it sets up the equation to be squared, eliminating the complex square roots.

Isolating variables correctly is the first step towards simplifying the equation and finding any potential solutions, leading to a clearer path to checking their validity.