Problem 47 - A Graphical Approach to College Algebra

Question

Use an analytic method to solve each equation in part (a). Support the solution with a graph. Then use the graph to solve the inequalities in parts (b) and (c). (a) \(\left(x^{2}+6 x\right)^{1 / 4}=2\) (b) \(\left(x^{2}+6 x\right)^{1 / 4}>2\) (c) \(\left(x^{2}+6 x\right)^{1 / 4}<2\)

Step-by-Step Solution

Answer

(a) \( x = -8 \) or \( x = 2 \); (b) \( x < -8 \) or \( x > 2 \); (c) \( -8 < x < 2 \).

Step 1: Solve the Equation Analytically

To solve \( \left(x^2 + 6x\right)^{1/4} = 2 \), start by eliminating the fourth root by raising both sides of the equation to the power of 4. This gives us \( x^2 + 6x = 2^4 \). Simplify the right-hand side to get \( x^2 + 6x = 16 \). Next, rearrange the equation to form a standard quadratic equation: \( x^2 + 6x - 16 = 0 \). Now, solve for \( x \) using the quadratic formula: \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), where \( a = 1 \), \( b = 6 \), and \( c = -16 \). Substitute these values into the formula to find the roots. Calculate the discriminant \( b^2 - 4ac \): \( 6^2 - 4 \times 1 \times (-16) = 36 + 64 = 100 \). Thus, the roots are \( x = \frac{-6 \pm \sqrt{100}}{2} = \frac{-6 \pm 10}{2} \), which results in \( x_1 = 2 \) and \( x_2 = -8 \).

Step 2: Graph the Function

Graph the function \( y = \left(x^2 + 6x\right)^{1/4} \) alongside the line \( y = 2 \). This will provide a visual representation of the solution, which ensures that the analytic solutions \( x = 2 \) and \( x = -8 \) are correct as they are the points where the graph of the function intersects the line \( y = 2 \).

Step 3: Solve the Inequality \( (a) \) with the Graph

For the inequality \( \left(x^2 + 6x\right)^{1/4} > 2 \), refer to the graph. Look above the line \( y = 2 \) to determine the values of \( x \). These values are for \( x \) less than -8 and for \( x \) greater than 2. Therefore, the solution set for (b) is \( x < -8 \) or \( x > 2 \).

Step 4: Solve the Inequality \( (b) \) with the Graph

To solve \( \left(x^2 + 6x\right)^{1/4} < 2 \), observe the portions of the graph that fall below the \( y = 2 \) line. The values of \( x \) that meet this criterion are between -8 and 2. Therefore, the solution set for (c) is \( -8 < x < 2 \).

Key Concepts

Quadratic Formula

The quadratic formula is a powerful tool for solving any quadratic equation of the form \( ax^2 + bx + c = 0 \). By using this formula, we can find the roots of the equation without having to factor manually. The formula is given by:
\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]
In our exercise, after simplifying the original equation to \( x^2 + 6x - 16 = 0 \), the coefficients are \( a = 1 \), \( b = 6 \), and \( c = -16 \). To use the quadratic formula:

• First, calculate the discriminant \( b^2 - 4ac \).
• If the discriminant (\( 100 \) in this case) is a positive perfect square, it indicates two real and distinct solutions.
• The solutions here are \( x_1 = 2 \) and \( x_2 = -8 \).

Understanding how to apply the quadratic formula not only finds solutions but also helps in verifying with additional methods.

Graphical Solutions

Graphing provides a visual method to verify the solutions of equations and inequalities. To interpret our solutions graphically, we need to plot:

• The function \( y = \left(x^2 + 6x\right)^{1/4} \)
• The line \( y = 2 \)

Where these plots intersect, the x-values correspond to the solutions of the equation. In this scenario, intersections at \( x = 2 \) and \( x = -8 \) confirm our analytical solutions.
This approach also aids in understanding inequalities. For \( \left(x^2 + 6x\right)^{1/4} > 2 \), identify x-values where the function is above the line. For \( \left(x^2 + 6x\right)^{1/4} < 2 \), it's where the function is below. Graphical solutions provide an intuitive perspective, ensuring our analytical results are accurate.

Inequalities

Inequalities involve finding the range of values that satisfy a given condition. In this exercise, we have to consider the inequalities \( \left(x^2 + 6x\right)^{1/4} > 2 \) and \( \left(x^2 + 6x\right)^{1/4} < 2 \). The steps are:

• For \( \left(x^2 + 6x\right)^{1/4} > 2 \), the solution consists of any x-values where the graph is above the line, giving us \( x < -8 \) or \( x > 2 \).
• Conversely, for \( \left(x^2 + 6x\right)^{1/4} < 2 \), we look for x-values below the line, resulting in \( -8 < x < 2 \).

Visualizing these inequalities on a graph allows students to easily grasp the solution regions. This helps in understanding complex problems effortlessly.

Analytic Methods

Analytic methods for solving equations involve algebraic manipulation to simplify expressions and isolate unknowns. In essence, these methods are purely algebraic and do not rely on visual aspects.
For our given example:

• We removed the fourth root by raising both sides to the power of four, simplifying the task to solving a quadratic equation.
• This led directly to forming a standard quadratic equation, which could be solved using the quadratic formula.

Analytical approaches are beneficial when graphical methods might be too complex or imprecise. They provide a clear, exact solution through systematic steps, reinforcing mathematical rigor and precision.