Problem 47 - A Graphical Approach to Precalculus with Limits

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) Solutions: \( x = 2 \) and \( x = -8 \). (b) \( x < -8 \) or \( x > 2 \). (c) \( -8 < x < 2 \).

Step 1: Solve the Equation

To solve the equation \( \left(x^{2}+6x\right)^{1/4}=2 \), we start by raising both sides to the fourth power to eliminate the fractional exponent, leading to the equation \( x^{2} + 6x = 2^4 = 16 \).

Step 2: Rearrange the Equation

Rearrange the equation to standard quadratic form: \( x^{2} + 6x - 16 = 0 \).

Step 3: Use the Quadratic Formula

Use the quadratic formula, \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), where \( a=1, b=6, c=-16 \). Compute the discriminant: \( b^2 - 4ac = 36 + 64 = 100 \).

Step 4: Calculate the Roots

The roots are \( x = \frac{-6 \pm 10}{2} \). Thus, the solutions are \( x = 2 \) and \( x = -8 \).

Step 5: Graphing the Function

Graph the function \( y = \left(x^{2}+6x\right)^{1/4} \) along with the line \( y = 2 \). The points of intersection at \( x = 2 \) and \( x = -8 \) confirm our algebraic solutions.

Step 6: Solve the Inequality for (b)

For \(\left(x^{2}+6x\right)^{1/4} > 2 \), we solve \( x^{2} + 6x > 16 \). The quadratic \( x^{2} + 6x - 16 = 0 \) gives roots at \( x = -8 \) and \( x = 2 \). The expression is positive outside of these roots, leading to solutions \( x < -8 \) or \( x > 2 \).

Step 7: Solve the Inequality for (c)

For \(\left(x^{2}+6x\right)^{1/4} < 2 \), solve \( x^{2} + 6x < 16 \). The quadratic form \( x^{2} + 6x - 16 = 0 \) has roots at \( x = -8 \) and \( x = 2 \). The expression is negative between these roots, resulting in solutions \( -8 < x < 2 \).

Key Concepts

Quadratic Equations

Quadratic equations come in the standard form \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \) are constants. Solving these equations can often seem tricky, but with the use of formulas and systematic methods, it becomes manageable.

To solve such equations, one common method is the quadratic formula:

• \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)

This formula is handy as it gives you the roots of the quadratic equation directly.

Before applying the quadratic formula, it's helpful to understand the term "discriminant," which is \( b^2 - 4ac \). This value helps determine the nature of the roots:

• If the discriminant is greater than zero, you'll have two distinct real roots.
• If it's equal to zero, there's exactly one real root, meaning the parabola touches the x-axis at one point.
• A negative value means the roots are complex, indicating the graph doesn't intersect the x-axis.

Recognizing these characteristics can assist in visualizing and solving quadratic equations effectively.

Inequalities

Inequalities seem similar to equations but tell us more about a range of values rather than specific points. Different types of inequalities exist, such as \( < \), \( > \), \( \leq \), and \( \geq \). Solving inequalities involves finding these ranges and understanding how they relate to equations. For instance, when solving \( x^2 + 6x > 16 \) derived from the equation after transposing terms, you must find where the expression is positive or negative.

To solve for inequalities, follow these key steps:

• Solve the related equation \( x^2 + 6x = 16 \) to find critical points where the expression equals zero, giving boundary values.
• An analysis of the intervals determined by these points helps classify where the original inequality holds by checking values in these intervals.
• Another useful way is graphing the quadratic function and observing where it is above or below a certain horizontal line, like \( y = 16 \).

These steps aid in judging where the inequalities hold true, revealing the solution set as ranges along the number line.

Graphing Functions

Graphing functions is an essential skill that provides visual insight into mathematical expressions, making problem-solving and analysis easier. In our exercise, graphing helps verify solutions to quadratic equations and inequalities. For example, the graph of a function like \( y = (x^2 + 6x)^{1/4} \) along with a line \( y = 2 \) allows us to observe real intersections, confirming computed answers.

Here's a straightforward guide to graphing:

• Start by plotting any known points, such as roots obtained from solving equations.
• Sketch the general shape of the function based on its characteristics (J-shaped, U-shaped, etc.). For quadratics, knowing whether it opens upwards or downwards is determined by the sign of \( a \) in \( ax^2 + bx + c \).
• Add any horizontal lines or other function values related to inequalities. These help determine where the function's value crosses certain thresholds.

Coupling algebraic solutions with graphing paints a fuller picture of a problem, aiding in visual learning and better understanding of the nature of equations and inequalities.