Discriminant Analysis Determine Solutions Of Quadratic Equation 2x² + 5x - 8 = 0

In the realm of mathematics, quadratic equations hold a prominent position. These equations, characterized by their highest power being two, exhibit a rich variety of solutions. One powerful tool for deciphering the nature of these solutions is the discriminant. In this comprehensive guide, we will delve into the concept of the discriminant and explore how it empowers us to determine the number and type of solutions for a given quadratic equation. Specifically, we will focus on the equation 2x² + 5x - 8 = 0, illustrating the application of the discriminant in a step-by-step manner.

What is a Quadratic Equation?

Before we delve into the discriminant, let's briefly revisit the concept of a quadratic equation. A quadratic equation is a polynomial equation of the second degree, generally expressed in the standard form:

ax² + bx + c = 0

where a, b, and c are constants, and x represents the variable we aim to solve for. The solutions to a quadratic equation are also known as its roots or zeros, representing the values of x that satisfy the equation. These roots correspond to the points where the parabola represented by the quadratic equation intersects the x-axis on a graph.

Understanding quadratic equations is crucial in various fields, including physics, engineering, and economics, where they model numerous real-world phenomena, such as projectile motion, electrical circuits, and optimization problems.

The Discriminant: Unveiling the Nature of Solutions

The discriminant is a crucial component of the quadratic formula, providing valuable insights into the nature of the solutions of a quadratic equation. The quadratic formula, a cornerstone for solving quadratic equations, is given by:

x = (-b ± √(b² - 4ac)) / 2a

The discriminant, denoted by Δ (Delta), is the expression under the square root in the quadratic formula:

Δ = b² - 4ac

The discriminant holds the key to understanding the number and type of solutions a quadratic equation possesses. By analyzing the discriminant's value, we can determine whether the equation has two distinct real solutions, one real solution (a repeated root), or two complex solutions.

Interpreting the Discriminant

The discriminant can take on three possible values:

• Δ > 0: If the discriminant is positive, the quadratic equation has two distinct real solutions. This means the parabola intersects the x-axis at two different points.
• Δ = 0: If the discriminant is zero, the quadratic equation has exactly one real solution (a repeated root). In this case, the parabola touches the x-axis at only one point, representing the vertex of the parabola.
• Δ < 0: If the discriminant is negative, the quadratic equation has two complex solutions (conjugate pairs). This signifies that the parabola does not intersect the x-axis at any real points.

Applying the Discriminant to Our Example: 2x² + 5x - 8 = 0

Now, let's put our knowledge of the discriminant into action by analyzing the quadratic equation 2x² + 5x - 8 = 0. Our goal is to determine the number and type of solutions this equation possesses.

Identifying the Coefficients

First, we need to identify the coefficients a, b, and c in our equation. Comparing our equation to the standard form ax² + bx + c = 0, we can readily identify the coefficients:

• a = 2
• b = 5
• c = -8

Calculating the Discriminant

Next, we will calculate the discriminant using the formula Δ = b² - 4ac. Substituting the values of a, b, and c, we get:

Δ = (5)² - 4(2)(-8)

Δ = 25 + 64

Δ = 89

Analyzing the Discriminant's Value

Our calculated discriminant, Δ = 89, is a positive value. Based on our understanding of the discriminant's interpretation, we can conclude that the quadratic equation 2x² + 5x - 8 = 0 has two distinct real solutions.

Finding the Solutions Using the Quadratic Formula

To find the actual solutions, we can now employ the quadratic formula:

x = (-b ± √(b² - 4ac)) / 2a

Substituting the values of a, b, c, and Δ, we get:

x = (-5 ± √89) / (2 * 2)

x = (-5 ± √89) / 4

Thus, the two distinct real solutions for the equation are:

x₁ = (-5 + √89) / 4

x₂ = (-5 - √89) / 4

These solutions represent the x-coordinates of the points where the parabola defined by the equation 2x² + 5x - 8 = 0 intersects the x-axis.

Conclusion: The Power of the Discriminant

The discriminant stands as a powerful tool in the realm of quadratic equations. By calculating the discriminant, we can swiftly determine the number and type of solutions a quadratic equation possesses without resorting to the full quadratic formula. This understanding empowers us to gain valuable insights into the behavior of quadratic functions and their applications in various mathematical and real-world contexts.

In this comprehensive guide, we have explored the concept of the discriminant, its interpretation, and its application to the quadratic equation 2x² + 5x - 8 = 0. Through this example, we have demonstrated how the discriminant allows us to efficiently determine that the equation has two distinct real solutions. Mastering the discriminant equips you with a valuable tool for navigating the world of quadratic equations and their diverse applications.

Further Exploration

To deepen your understanding of quadratic equations and the discriminant, consider exploring the following topics:

• Graphing Quadratic Equations: Visualize the relationship between the discriminant and the parabola's intersection with the x-axis.
• Applications of Quadratic Equations: Discover real-world scenarios where quadratic equations are used, such as projectile motion and optimization problems.
• Completing the Square: Learn an alternative method for solving quadratic equations.
• Complex Numbers: Explore the realm of complex solutions for quadratic equations with negative discriminants.

By delving into these topics, you will gain a more comprehensive understanding of quadratic equations and their significance in mathematics and beyond.