Problem 54 - Algebra and Trigonometry Real Mathematics, Real People

Question

Use a graphing utility to determine the number of real solutions of the quadratic equation. $$\frac{1}{3} x^{2}-5 x+25=0$$

Step-by-Step Solution

Answer

The number of intersections with the x-axis on the graph of the quadratic equation will give the number of real solutions of the given equation. This can best be viewed and solved by using a graphing utility. The actual number of solutions may vary based on the graph.

Step 1: Understanding the Quadratic Equation Graph Shape

Start by understanding the graph shape of a quadratic equation. The graph of any quadratic equation is a parabola. If the coefficient of the \(x^{2}\) term is positive, the parabola opens upwards, and if it is negative, the parabola opens downwards. In our given equation, the coefficient of the \(x^{2}\) term is positive (\(\frac{1}{3}\)), so it is clear that our graph will open upwards.

Step 2: Graphing the Quadratic Equation

Using a graphing utility of your choice (like Desmos, GeoGebra, etc.), graph the equation \(\frac{1}{3} x^{2}-5 x+25=0\). Simply input the function and observe the resulting graph.

Step 3: Counting the Number of Real Solutions

Looking at the graph of the quadratic equation, the real solutions are the x-values where the graph intersects the x-axis. Count how many times the graph intersects the x-axis to find the number of real roots of the quadratic equation.

Key Concepts

Understanding Parabolas

Let's start by understanding the fundamental shape in quadratic equations: the parabola. A parabola is a U-shaped curve that is symmetrical and can open either upwards or downwards. The direction in which a parabola opens is determined by the coefficient of the quadratic term \(x^2\).

• If the coefficient of \(x^2\) is positive, the parabola will open upwards.
• If the coefficient is negative, the parabola will open downwards.

In the equation \(\frac{1}{3} x^{2} - 5x + 25 = 0\), the quadratic term \(\frac{1}{3}\) is positive. Therefore, the graph of this equation, or the parabola, opens upwards. Visualizing this shape is crucial as it sets the stage for determining other characteristics of the equation, like the vertex and the roots.

Using Graphing Utilities Effectively

To fully explore quadratic equations, it's helpful to use graphing utilities. Tools like Desmos or GeoGebra allow you to input an equation and immediately visualize the graph. This is especially useful for determining the nature of the solutions without manually solving the equation.When graphing the equation \(\frac{1}{3} x^{2} - 5x + 25 = 0\), type the expression into the graphing tool and observe how the graph appears. These utilities often offer features to zoom in and out, or move the graph around. This capability can help you closely analyze where the parabola intersects the x-axis, which is key for identifying real solutions.Using graphing utilities is a skill in itself. Familiarizing yourself with features such as plotting points, adjusting the window, and analyzing intersections can make solving quadratic equations much easier and quicker.

Counting Real Solutions on the Graph

In the context of quadratic equations, real solutions correspond to the points where the parabola intersects the x-axis. These intersections indicate where the value of the function is zero, revealing the roots of the equation.For the equation \(\frac{1}{3} x^{2} - 5x + 25 = 0\), after plotting the graph using a utility, you need to count how many times the parabola crosses the x-axis.

• If the parabola intersects the x-axis twice, the equation has two distinct real solutions.
• If it only touches the x-axis at one point, there is one real solution, known as a repeated root.
• If the parabola does not touch the x-axis, there are no real solutions, meaning the roots are complex or imaginary.

For this specific equation, you might find that the parabola doesn't intersect the x-axis at all. This indicates that the equation has no real solutions, providing a clear graphical representation of its roots.