Determine Real-Number Solutions Of Quadratic Function Using The Discriminant

In the realm of mathematics, quadratic functions hold a significant place, and understanding their properties is crucial for various applications. One essential aspect of quadratic functions is determining the number of real-number solutions they possess. This article delves into the methods to ascertain the number of real-number solutions for a given quadratic function, with a particular focus on utilizing the discriminant. Let's consider the quadratic function: $y = -7x^2 + 9x - 5$.

Understanding Quadratic Functions and Solutions

Before we dive into the specifics, let's establish a solid understanding of quadratic functions and what constitutes a solution. A quadratic function is a polynomial function of degree two, generally expressed in the form $ax^2 + bx + c = 0$, where a, b, and c are constants, and a ≠ 0. The solutions, also known as roots or zeros, of a quadratic function are the values of x that satisfy the equation, making the function equal to zero. Graphically, these solutions represent the points where the parabola intersects the x-axis.

Real-number solutions are solutions that belong to the set of real numbers, which encompasses all rational and irrational numbers. Quadratic functions can have two distinct real solutions, one real solution (a repeated root), or no real solutions. The nature of the solutions is dictated by a critical component called the discriminant.

The Discriminant: Unveiling the Nature of Solutions

The discriminant is a powerful tool for determining the number and type of solutions a quadratic equation possesses. It is derived from the quadratic formula, which provides the solutions for any quadratic equation in the form $ax^2 + bx + c = 0$:

$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

The discriminant, denoted as Δ, is the expression under the square root in the quadratic formula:

$Δ = b^2 - 4ac$

The discriminant's value reveals the following information about the solutions:

• If Δ > 0: The quadratic equation has two distinct real solutions. This means the parabola intersects the x-axis at two different points.
• If Δ = 0: The quadratic equation has one real solution (a repeated root). The parabola touches the x-axis at exactly one point.
• If Δ < 0: The quadratic equation has no real solutions. The parabola does not intersect the x-axis.

Applying the Discriminant to Our Example

Now, let's apply the discriminant to the given quadratic function: $y = -7x^2 + 9x - 5$. To use the discriminant, we first identify the coefficients a, b, and c:

• a = -7
• b = 9
• c = -5

Next, we substitute these values into the discriminant formula:

$Δ = b^2 - 4ac = (9)^2 - 4(-7)(-5)$

Calculating the discriminant:

$Δ = 81 - 140 = -59$

Since the discriminant (Δ = -59) is negative, we can conclude that the quadratic function has no real solutions.

Interpreting the Result

The negative discriminant signifies that the parabola represented by the quadratic function does not intersect the x-axis. This means there are no real values of x that will make the function equal to zero. The solutions, in this case, are complex numbers, which involve the imaginary unit i (where i² = -1).

In graphical terms, the parabola either lies entirely above or entirely below the x-axis, depending on the sign of the leading coefficient (a). In our example, a = -7, which is negative, indicating that the parabola opens downwards and lies entirely below the x-axis.

Steps to Determine the Number of Real-Number Solutions Using the Discriminant

To summarize, here's a step-by-step guide to determining the number of real-number solutions for a quadratic function using the discriminant:

1. Identify the coefficients: Write the quadratic function in the standard form $ax^2 + bx + c = 0$ and identify the values of a, b, and c.
2. Calculate the discriminant: Substitute the values of a, b, and c into the discriminant formula: $Δ = b^2 - 4ac$.
3. Interpret the discriminant:
   • If Δ > 0, there are two distinct real solutions.
   • If Δ = 0, there is one real solution (a repeated root).
   • If Δ < 0, there are no real solutions.

Examples of Different Cases

To further illustrate the concept, let's consider a few more examples:

Example 1: Two Real Solutions

Consider the quadratic function $y = x^2 - 5x + 6$. Here, a = 1, b = -5, and c = 6. Calculating the discriminant:

$Δ = (-5)^2 - 4(1)(6) = 25 - 24 = 1$

Since Δ > 0, this quadratic function has two distinct real solutions. These solutions can be found by factoring the quadratic as (x - 2)(x - 3) = 0, which gives x = 2 and x = 3.

Example 2: One Real Solution

Consider the quadratic function $y = x^2 - 4x + 4$. Here, a = 1, b = -4, and c = 4. Calculating the discriminant:

$Δ = (-4)^2 - 4(1)(4) = 16 - 16 = 0$

Since Δ = 0, this quadratic function has one real solution. This solution can be found by factoring the quadratic as (x - 2)² = 0, which gives x = 2.

Example 3: No Real Solutions

We already analyzed this case with $y = -7x^2 + 9x - 5$, where we found that Δ < 0, indicating no real solutions.

Importance of Understanding the Discriminant

The discriminant is more than just a mathematical formula; it's a gateway to understanding the behavior and characteristics of quadratic functions. By analyzing the discriminant, we can efficiently determine the number of real-number solutions without having to solve the quadratic equation explicitly.

This knowledge is valuable in various fields, including physics, engineering, and economics, where quadratic functions are used to model real-world phenomena. For example, in projectile motion, the discriminant can help determine whether a projectile will hit a target or not. In optimization problems, it can help identify the existence of maximum or minimum values.

Conclusion

Determining the number of real-number solutions for a quadratic function is a fundamental skill in mathematics. By utilizing the discriminant, we can quickly and accurately assess the nature of the solutions without the need for complex calculations. In the case of the quadratic function $y = -7x^2 + 9x - 5$, the discriminant revealed that it has no real solutions, highlighting the power and efficiency of this mathematical tool. Understanding the discriminant not only enhances our problem-solving abilities but also deepens our appreciation for the elegance and interconnectedness of mathematical concepts. Mastering this concept opens doors to more advanced topics and applications, making it an invaluable asset in any mathematical journey.

In summary, the discriminant $b^2 - 4ac$ is the key to unlocking the nature of solutions for a quadratic equation. For the given equation, $y = -7x^2 + 9x - 5$, the discriminant is -59, indicating no real-number solutions. This tool allows us to efficiently analyze quadratic functions and their behavior in various contexts.