Problem 64 - Introductory Algebra for College Students

Question

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. There is no relationship between the graph of \(y=a x^{2}+b x+c\) and the number of real solutions of the equation \(a x^{2}+b x+c=0\)

Step-by-Step Solution

Answer

The given statement is false. A correct version of the statement would be: There is a relationship between the graph of \(y=ax^2 + bx + c\) and the number of real solutions of the equation \(ax^2 + bx + c = 0\), determined by the discriminant.

Step 1: Understanding Quadratic Equation and its Solutions

A quadratic equation in standard form is \(ax^2+bx+c=0\) where \(a\), \(b\), and \(c\) are constants and \(a\) is nonzero. The solutions of this quadratic equation depends on the discriminant, \(b^2 - 4ac\).

Step 2: Understanding the Discriminant

The discriminant determines the solutions of a quadratic equation. \n\n1) If the discriminant is positive, the quadratic equation has two distinct real solutions. \n2) If the discriminant is zero, the quadratic equation has one real solution. \n3) If the discriminant is negative, the quadratic equation has no real solutions or two complex solutions.

Step 3: Relating the Discriminant to the Graph of the equation

The discriminant affects the graph of the quadratic equation in the following ways: \n\n1) If the discriminant is positive, the graph of the equation will intersect the x-axis at two points. \n2) If the discriminant is zero, the graph of the equation will touch the x-axis at a single point, called the vertex. \n3) If the discriminant is negative, the graph will not intersect or touch the x-axis at all.

Step 4: Evaluating the Given Statement

Now that we have an understanding of the relationship between the discriminant and the graph of the equation, we can evaluate the given statement. The statement says that there is no relationship between the graph of \(y=ax^2 + bx + c\) and the number of real solutions of \(ax^2 +bx + c = 0\). This statement is false; as we have established, the graph and the number of real solutions are directly related through the discriminant.

Step 5: Correcting the Statement

A correct version of the statement would be: 'There is a relationship between the graph of \(y=ax^2 + bx + c\) and the number of real solutions of the equation \(ax^2 + bx + c = 0\).' This relationship is determined by the discriminant, \(b^2 - 4ac\).

Key Concepts

Discriminant

The discriminant is a crucial concept in understanding quadratic equations. In the standard quadratic equation format, \(ax^2 + bx + c = 0\), the discriminant is found using the formula \(b^2 - 4ac\). This value helps us determine the nature and number of solutions the equation has. Let's break down how it works:

• A positive discriminant \((b^2 - 4ac > 0)\) indicates that the quadratic equation has two distinct real solutions.
• A discriminant of zero \((b^2 - 4ac = 0)\) shows that there is exactly one real solution, often called a repeated or double root.
• A negative discriminant \((b^2 - 4ac < 0)\) reveals that there are no real solutions; instead, the solutions are complex numbers.

Understanding the discriminant provides insight into the nature of the solutions to the quadratic equation without actually solving it, making it an efficient tool in solving quadratic problems.

Real Solutions

Real solutions refer to the values of \(x\) that satisfy the equation \(ax^2 + bx + c = 0\) and are real numbers. The discriminant plays a key role in determining how many real solutions there are:

- **Two distinct real solutions:** This occurs when the discriminant is positive. This means that there are two different \(x\) values satisfying the equation. They correspond to the points where the graph of the quadratic equation intersects the x-axis.- **One real solution:** When the discriminant equals zero, the quadratic has exactly one real solution. This single solution represents the vertex of the parabola touching the x-axis.- **No real solutions:** With a negative discriminant, the quadratic equation has no real solutions. Instead, the solutions are complex numbers, indicating the graph does not touch or intersect the x-axis.

Identifying the number of real solutions is important for graphing and solving quadratic equations and helps visualize the solutions' locations on the graph.

Graph of Quadratic

The graph of a quadratic equation \(y = ax^2 + bx + c\) is a parabola. The relationship between the quadratic equation's graph and its solutions is visibly evident on the x-axis.

- **Intersections with the x-axis:** If a quadratic equation has real solutions, they correspond to the points where the graph intersects the x-axis. With a positive discriminant, these intersections occur at two distinct points, while a zero discriminant means the parabola just touches the x-axis at one point (the vertex).- **No Intersection:** When the discriminant is negative, there is no x-axis intersection, implying no real solutions. The parabola lies entirely above or below the x-axis depending on the leading coefficient \(a\).- **Shape and Direction:** Though the discriminant doesn't directly affect the parabola's shape, the sign of \(a\) does. A positive \(a\) opens the parabola upwards, while a negative \(a\) opens it downwards.

The graph of a quadratic equation is a handy visual tool to quickly assess the nature of its solutions and understand the real-world implications of the equation.