Problem 53 - A Graphical Approach to College Algebra

Question

Use a graphical method to find all real solutions of each equation. Express solutions to the nearest hundredth. $$2.45 x^{4}-3.22 x^{3}=-0.47 x^{2}+6.54 x+3$$

Step-by-Step Solution

Answer

Real solutions are approximately: \(-1.34, 0.78, 2.29\).

Step 1: Rearrange the Equation

First, we move all terms to one side of the equation to set it equal to zero. This results in the equation: \[2.45x^4 - 3.22x^3 + 0.47x^2 - 6.54x - 3 = 0\]

Step 2: Plot the Function

Graph the function \(f(x) = 2.45x^4 - 3.22x^3 + 0.47x^2 - 6.54x - 3\) using graphing software or a graphing calculator.

Step 3: Identify X-intercepts

Observe the graph and identify where it intersects the x-axis. These x-intercepts represent the real solutions to the equation \(f(x) = 0\).

Step 4: Approximate the Solutions

Using the graph, approximate the x-values of the intercepts to the nearest hundredth. Ensure that the graph is zoomed appropriately to allow for accurate reading of the intercept values.

Key Concepts

Polynomial Equations

Polynomial equations are algebraic expressions made up of variables raised to whole number powers, constants, and coefficients. They can range from simple linear equations to more complex ones involving higher power terms such as quadratic, cubic, or quartic (fourth degree) equations. In the given problem, the polynomial equation is of the fourth degree, written in standard form as follows:

• \(2.45x^4 - 3.22x^3 + 0.47x^2 - 6.54x - 3 = 0\)

Understanding these terms:

• **Degree:** The highest power of the variable; in this case, it is 4, indicating a quartic polynomial.
• **Coefficients:** Numbers multiplying the powers of \(x\), such as 2.45, -3.22, etc.
• **Constants:** Terms without a variable, here it's the number -3.

Polynomials can have multiple real solutions based on their degree. Typically, a fourth-degree polynomial might have up to four real roots, though these can sometimes be complex as well. Graphical methods give a visual approach to finding these real solutions without dealing with complex algebraic manipulations.

X-Intercepts

X-intercepts are points where a graph crosses the x-axis. For mathematical functions, these intercepts represent the solutions to the equation set to zero. In algebra terms, x-intercepts are values of \(x\) for which \(f(x) = 0\). In this exercise, finding the x-intercepts of the quartic polynomial helps determine its real solutions.

• **Why are X-Intercepts Important?**: They provide a straightforward way to find solutions visually.
• **How to Identify X-Intercepts on a Graph?**: Look for points where the curve of the graph meets the horizontal axis (x-axis).
• **Using Graphing Tools**: Tools like graphing calculators or graphing software make it easier to visualize and pinpoint these intercepts with accuracy.

Once identified on the graph, the x-intercepts give the real solutions to the polynomial equation. In our problem's graph of \(f(x) = 2.45x^4 - 3.22x^3 + 0.47x^2 - 6.54x - 3\), these are the points of interest for solutions.

Real Solutions

Real solutions of polynomial equations are values of the variable that satisfy the equation, making it true. These are essentially the x-values of the x-intercepts seen on a graph. Unlike complex solutions, real solutions are numbers that can be plotted on the traditional number line and visualized on the coordinate plane.
Identifying real solutions graphically involves:

• **Analyzing the Graph**: Look for intersections with the x-axis, which correspond to the real roots.
• **Precision**: With tools, ensure the graph is properly zoomed to read the x-intercepts accurately. Approximations like nearest hundredth are necessary when the intercepts aren't exact whole numbers.
• **Multiple Roots**: A polynomial of degree four can possibly have up to four x-intercepts, but may have fewer depending on the nature of each root (real or complex).

Graphical methods provide a useful, intuitive way to approximate real solutions, bypassing the need for complex calculations, especially when polynomial equations involve higher degrees or complicated coefficients.