Problem 47 - Algebra 1

Question

Solve the equation. Check for extraneous solutions. $$\frac{2}{3} x=\sqrt{24 x-128}$$

Step-by-Step Solution

Answer

Following these steps will provide the solutions for x. Those solutions must be checked in the original equation to avoid extraneous solutions. If any solution does not hold true in the original equation, it is an extraneous solution and must be discarded.

Step 1: Square both sides of the equation

To get rid of the radical, square both sides of the equation \(\frac{2}{3} x\)^2 = (\sqrt{24x-128})^2. Simplifying the equation produces \frac{4}{9} x^{2} = 24x -128.

Step 2: Simplify the equation to a standard form

Subtract both sides by 24x and add 128 to achieve a standard quadratic form. This yields \frac{4}{9} x^{2} - 24x + 128 = 0.

Step 3: Solve the quadratic equation

Since this is a quadratic equation, try to solve it using the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\). In this case, a = \frac{4}{9}, b = -24, and c = 128. Substituting these values, we get two potential solutions for x.

Step 4: Check for extraneous solutions

Plug both of these solutions back into the original equation to check if they are valid. Discard any solutions that do not hold true in the original equation.

Key Concepts

Radicals and Rational Exponents

Understanding radicals and rational exponents is vital when solving equations that involve roots and powers. A radical, often represented as \( \sqrt[n]{x} \), is essentially an expression that denotes the nth root of a number x. When we square both sides of an equation to eliminate a square root, we're leveraging the property that \( (\sqrt[n]{x})^2 = x \), assuming x is non-negative.

Rational exponents, on the other hand, provide a different but equivalent way to express roots using fractions. For example, \( x^{\frac{1}{n}} \) is the same as \( \sqrt[n]{x} \) and \( x^{\frac{m}{n}} \) is equivalent to \( (\sqrt[n]{x})^m \). This correspondence is useful because it allows us to apply the rules of exponents, like multiplication and division, to simplify expressions involving radicals.

During the process of solving the given exercise, squaring both sides is the method applied to remove the square root. However, while squaring gets rid of the radical, it can introduce extraneous solutions. Therefore, after applying such operations, it's critical to verify the solutions by plugging them back into the original equation.

Quadratic Formula

The quadratic formula is a powerful tool for solving quadratic equations of the form \( ax^2 + bx + c = 0 \). The formula is given by \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \). It directly provides the solutions for x by using the coefficients a, b, and c from the equation.

Using the quadratic formula involves several steps: identifying the coefficients (a, b, and c), computing the discriminant (\( b^2 - 4ac \)), which determines the nature of the roots, and then applying these values within the formula to find the solutions for x. It's an invaluable method when factoring is not straightforward or possible.

In the given exercise, after transforming the equation to standard quadratic form, we used the quadratic formula to solve for x. It's critical to properly substitute the coefficients correctly, including their signs, to ensure accurate computation of solutions.

Extraneous Solutions

Extraneous solutions are results that emerge from the algebraic process of solving equations but are not true solutions to the original equation. These often occur when both sides of an equation are manipulated in ways that are not reversible, such as squaring both sides of an equation containing a radical.

To check for extraneous solutions, each potential solution must be substituted back into the original equation. If the substitution results in a true statement, then the solution is valid. However, if it leads to false statements, it is deemed as an extraneous solution and must be discarded.

In the exercise, after applying the quadratic formula, we arrive at two potential solutions. The last step requires us to verify these solutions in the original equation, thus filtering out the extraneous ones. This step is crucial and ensures that all reported solutions are legitimate and applicable to the context of the problem.