Problem 53 - Algebra and Trigonometry Real Mathematics, Real People

Question

Use a graphing utility to determine the number of real solutions of the quadratic equation. $$\frac{4}{7} x^{2}-8 x+28=0$$

Step-by-Step Solution

Answer

After drawing the graph, you can see that the graph touches the x-axis at one point only. Therefore, the quadratic equation has only one real solution.

Step 1: Understand the Equation

Rewrite the equation. Given quadratic equation is \(\frac{4}{7} x^{2} - 8x + 28 = 0\) or \(x^{2} - 14x + 49 = 0\). In this rewritten form, the equation can be seen as a quadratic equation \(ax^{2}+bx+c = 0\) where \(a = 1\), \(b = -14\), and \(c = 49\).

Step 2: Draw the Graph

Plot the equation on the graphing utility. It's a simple parabola. Make sure the graph is drawn accurately.

Step 3: Identify the Roots

Identify the points at which the function intersects the x-axis. These points represent the solutions or roots of the equation.

Step 4: Count the solutions

Count the number of intersection points and that will be the number of real solutions. Remember, if the graph touches or intersects the x-axis, then it has real roots. Similarly, if the graph does not touch or intersect the x-axis, it has no real solutions.

Key Concepts

Real Solutions

When we talk about real solutions in the context of a quadratic equation, we mean the values of the variable, usually denoted by \( x \), that satisfy the equation when it equals zero. For a quadratic equation of the form \( ax^2 + bx + c = 0 \), real solutions are represented by the points where the graph of the equation intersects the x-axis.

These solutions can also be found using the discriminant \( b^2 - 4ac \). If the discriminant is positive, the quadratic has two distinct real solutions. If it is zero, the quadratic has exactly one real solution (also known as a repeated or double root). If the discriminant is negative, there are no real solutions.

Graphing Utility

A graphing utility is a tool, either software or a calculator, that allows us to visually plot equations, such as a quadratic equation, to understand their behavior. By inputting the equation into this tool, we can generate a graph that shows us where the solutions, or roots, might be.

Using a graphing utility for the quadratic equation \( \frac{4}{7}x^2 - 8x + 28 = 0 \), we observe the shape of a parabola on the graph. It helps in verifying our algebraic calculations by visually confirming the number and nature of the real solutions. The graphical method complements algebraic methods by providing an intuitive understanding of the solutions.

Parabola

In algebra, the graph of a quadratic equation \( ax^2 + bx + c = 0 \) is called a parabola. A parabola is a U-shaped curve that can open upwards or downwards depending on the sign of \( a \). If \( a \) is positive, the parabola opens upwards. If \( a \) is negative, it opens downwards.

For the given equation \( \frac{4}{7}x^2 - 8x + 28 = 0 \), the rewritten form as \( x^2 - 14x + 49 = 0 \) reveals a parabola that opens upwards. The vertex of this parabola represents the minimum point since the parabola opens upwards. This vertex will also provide the solution if it lies exactly on the x-axis.

Roots of the Equation

The roots of a quadratic equation are the values of \( x \) for which the equation equals zero. These roots correspond to the points where the parabola intersects the x-axis.
Identifying the roots is essential because:

• They tell us the x-values where the quadratic function has no output value (i.e., where it crosses the x-axis).
• They help solve real-world problems modeled by quadratic functions, such as calculating projectile motion or optimizing areas.

The roots can be found algebraically by factoring, using the quadratic formula \( x = \frac{-b \pm \sqrt{b^2-4ac}}{2a} \), or by completing the square. Using the graphing method, the roots are where the curve intersects the x-axis.