Problem 54 - A Graphical Approach to Precalculus with Limits

Question

Use a graphical method to find all real solutions of each equation. Express solutions to the nearest hundredth. $$\sqrt{17} x^{4}-\sqrt{22} x^{2}=-1$$

Step-by-Step Solution

Answer

The graphical solution shows approximate real solutions at the intersection points of \( f(x) \) and \( g(x) = -1 \).

Step 1: Define the Functions

We are given the equation \( \sqrt{17} x^4 - \sqrt{22} x^2 = -1 \). To use a graphical method, define two functions: \( f(x) = \sqrt{17} x^4 - \sqrt{22} x^2 \) and \( g(x) = -1 \). The task is to find where these two functions intersect, which represent the solutions to the equation.

Step 2: Graph the Functions

Next, plot the functions \( f(x) = \sqrt{17} x^4 - \sqrt{22} x^2 \) and \( g(x) = -1 \) on a graph to visualize their intersection points. The function \( f(x) \) is a quartic polynomial, while \( g(x) \) is a horizontal line. You can use graphing software or a graphing calculator for accurate plotting.

Step 3: Identify Intersection Points

On the graph, locate the points where the curve of \( f(x) \) intersects the line \( g(x) \). These intersection points correspond to the real solutions of the equation \( \sqrt{17} x^4 - \sqrt{22} x^2 = -1 \).

Step 4: Approximate Solutions

Use the graph to estimate the \( x \)-values of the intersection points to the nearest hundredth. Intersection points appear where the curve of \( f(x) \) crosses the line \( g(x) = -1 \).

Step 5: Verify Solutions

Substitute the estimated \( x \)-values back into the equation \( \sqrt{17} x^4 - \sqrt{22} x^2 \) to verify that they make it equal to \(-1\) to the nearest hundredth. Adjust the values slightly if necessary to get more accurate results.

Key Concepts

Real Solutions

In mathematics, real solutions refer to the values of the variable that satisfy a given equation, and they are found using various methods such as algebraic manipulation, numerical approximation, or graphical methods. When finding real solutions to an equation, it means we are looking for the points where the graph of the equation crosses the x-axis or intersects with another line or curve.

The real solutions are particularly significant because they provide the actual, tangible values of the variable that can be used in a real-world context. These solutions do not involve imaginary numbers, and they lie on the real number line. In the context of this exercise, the real solutions of the quartic polynomial equation are determined by locating the intersection points of two functions on a graph.

To find these real solutions using the graphical method explained in this exercise:

• First, define the functions involved.
• Graph the functions on the same axes.
• Identify the points where the graphs intersect to find the real solutions.
• Estimate the values to an appropriate precision, such as to the nearest hundredth.

Quartic Polynomial

A quartic polynomial is a polynomial of degree four, meaning that the highest power of the variable in the polynomial is four. More specifically, it takes the form \( ax^4 + bx^3 + cx^2 + dx + e = 0 \), where \( a, b, c, d, \) and \( e \) are constants, and \( a eq 0 \).

In the exercise, the quartic polynomial is given by the function \( f(x) = \sqrt{17} x^4 - \sqrt{22} x^2 \). This polynomial is identified as quartic due to the \( x^4 \) term. Solving quartic equations can be challenging using only algebraic methods, especially when an equation like this involves irrational numbers such as \( \sqrt{17} \) and \( \sqrt{22} \).

Quartic polynomials can have:

• Up to four real solutions.
• Complex solutions when there are fewer than four real solutions.

Using a graphical method simplifies finding the real solutions by allowing for visual identification of points where the equation satisfies specific conditions, such as equating to another line or function.

Intersection Points

Intersection points in the context of graphs represent the coordinates where two graphs meet or cross each other. These points are significant in mathematics as they often represent the solutions to an equation when two functions are equal.

For the exercise involving the quartic polynomial \( \sqrt{17} x^4 - \sqrt{22} x^2 = -1 \), intersection points are crucial to identify the real solutions. Here’s how it works:

• Graph the function \( f(x) = \sqrt{17} x^4 - \sqrt{22} x^2 \).
• Graph the horizontal line \( g(x) = -1 \).
• Where the graph of \( f(x) \) intersects the line \( g(x) \), those are the intersection points, giving the real solutions of the original equation.

Intersection points are visually identifiable, and by estimating their x-coordinates, you can determine the values of \( x \) that solve the equation to the nearest hundredth. Checking these points ensures they represent accurate real solutions that satisfy the equation's condition.