Problem 98 - Algebra 2

Question

Solve. Check for extraneous solutions. $$ \left(x^{2}-9\right)^{\frac{1}{2}}-x=-3 $$

Step-by-Step Solution

Answer

The solution, \(x = -3\), is an extraneous solution that does not validate the original equation. Thus, the given equation has no solution.

Step 1: Isolate the Square Root

Start by isolating the square root on one side of the equation by moving \(x\) to the other side: \(\sqrt{{x^2 - 9}} = x + 3\)

Step 2: Square Both Sides of the Equation

If you square both sides of the equation, the square root will be removed, leaving you with a quadratic equation: \(x^2 - 9 = x^2 + 6x + 9\)

Step 3: Simplify and Solve the Quadratic Equation

Subtract \(x^2\) from both sides of the equation to simplify then solve for \(x\): 0 = 6x + 18. Thus, \(x = -3\)

Step 4: Check for Extraneous Solutions

Substitute \(x = -3\) back into the original equation to check if it is valid: \(\sqrt{{(-3)^2 - 9}} + 3 = -3\). This simplifies to \(0 = -3\), which is not a true statement. So, \(x = -3\) is an extraneous solution.

Key Concepts

Quadratic Equation

Quadratic equations are fundamental in understanding various mathematical concepts. They are equations of the form \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are constants. In solving radical equations, such as the given exercise, it may be necessary to transform the equation into a quadratic form to find the roots or solutions.

When you isolate the square root on one side of the equation and then square both sides, you eliminate the square root, typically resulting in a quadratic equation. This quadratic form is crucial because it simplifies the problem, allowing you to use well-established methods like factoring or the quadratic formula to find potential solutions.

Isolating the Square Root

Isolating the square root is often the first step when solving an equation containing radicals. This concept means you rearrange the equation so that the square root expression stands alone on one side, helping to simplify further calculations.

In our exercise, the square root \(\sqrt{x^2 - 9}\) was initially part of a more complex expression. The equation was rewritten as \(\sqrt{x^2 - 9} = x + 3\), effectively isolating the square root. This step is crucial because it allows us to perform operations that eliminate the radical, specifically by squaring both sides.

• Isolate the square root expression on one side.
• Prepare for squaring, which simplifies the equation to a form that can more easily be solved.

Extraneous Solutions

Extraneous solutions are potential solutions that arise when solving equations, particularly during transformations, but do not satisfy the original equation.

In the process of solving radical equations, such transformations, like squaring both sides, can introduce these solutions. Unaided, you may conclude that \(x = -3\) is a valid solution after simplification. However, when checked against the original equation, it's clear that \(0 = -3\) is not valid, marking it as extraneous.

• Always verify potential solutions against the original equation.
• Understand that transformations can introduce solutions that are not applicable.

Checking Solutions

Checking solutions involves substituting them back into the original equation to ensure they hold true. This verification step is vital in confirming the validity of solutions, especially after possible changes during the solving process.

For the given exercise, after substituting \(x = -3\) into the original equation, we simplifed to find \(0 = -3\), a contradiction. This check shows that our solution is incorrect and extraneous.

• Substitute each solution back into the original equation.
• Confirm that the left and right sides of the equation are equal.
• Identify extraneous solutions by detecting contradictions or false statements.