Determine Solutions Of Quadratic Equation -24x² - 17x + 20 = 0

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Determining the nature of solutions for a quadratic equation is a fundamental concept in algebra. This article will guide you through the process of finding the type of solutions for the quadratic equation -24x² - 17x + 20 = 0. We'll explore the discriminant, which is a key tool in this analysis, and use it to classify the solutions as distinct rational, distinct irrational, or complex.

Understanding Quadratic Equations and Their Solutions

A quadratic equation is a polynomial equation of the second degree, generally expressed in the form ax² + bx + c = 0, where a, b, and c are constants, and a is not equal to zero. The solutions to a quadratic equation, also known as roots or zeros, are the values of x that satisfy the equation. These solutions represent the points where the parabola defined by the quadratic equation intersects the x-axis.

The Quadratic Formula: A Gateway to Solutions

The most common method for finding the solutions of a quadratic equation is the quadratic formula. This formula provides a direct way to calculate the roots, given the coefficients a, b, and c:

x = (-b ± √(b² - 4ac)) / 2a

This formula is derived by completing the square on the general form of the quadratic equation and is a cornerstone of algebra. Understanding and applying the quadratic formula is crucial for solving various mathematical problems, from simple algebraic manipulations to complex engineering calculations.

The Discriminant: Unveiling the Nature of Solutions

Within the quadratic formula lies a critical component called the discriminant, denoted as Δ (Delta). The discriminant is the expression under the square root:

Δ = b² - 4ac

The discriminant provides valuable information about the nature of the solutions without actually solving the equation. It tells us whether the solutions are real or complex, distinct or repeated, and rational or irrational. This powerful tool allows us to quickly classify the solutions and understand the behavior of the quadratic equation.

Analyzing the Discriminant: Real vs. Complex, Rational vs. Irrational

The value of the discriminant determines the type of solutions a quadratic equation possesses. Here’s a breakdown:

  1. Δ > 0 (Positive Discriminant): The equation has two distinct real solutions. This means the parabola intersects the x-axis at two different points. Furthermore:

    • If Δ is a perfect square, the solutions are rational. This indicates that the solutions can be expressed as a ratio of two integers.
    • If Δ is not a perfect square, the solutions are irrational. This implies that the solutions involve square roots that cannot be simplified to integers.

  2. Δ = 0 (Zero Discriminant): The equation has exactly one real solution (a repeated root). This means the parabola touches the x-axis at only one point, the vertex. The solution is always rational in this case.

  3. Δ < 0 (Negative Discriminant): The equation has two complex solutions (non-real solutions). This means the parabola does not intersect the x-axis. The solutions involve the imaginary unit i, where i² = -1.

Solving -24x² - 17x + 20 = 0: A Step-by-Step Approach

Now, let's apply this knowledge to the quadratic equation -24x² - 17x + 20 = 0. We will identify the coefficients, calculate the discriminant, and then classify the solutions.

Step 1: Identify the Coefficients

First, we need to identify the coefficients a, b, and c from the equation:

a = -24 b = -17 c = 20

These coefficients are the numerical values that define the quadratic equation. Correctly identifying them is crucial for the subsequent calculations and analysis.

Step 2: Calculate the Discriminant

Next, we calculate the discriminant using the formula Δ = b² - 4ac:

Δ = (-17)² - 4(-24)(20) Δ = 289 + 1920 Δ = 2209

The discriminant, Δ = 2209, is a positive number. This tells us that the quadratic equation has two distinct real solutions.

Step 3: Determine the Nature of the Solutions

Since Δ = 2209 is positive, we know there are two distinct real solutions. Now, we need to determine if these solutions are rational or irrational. To do this, we check if 2209 is a perfect square.

The square root of 2209 is 47 (√2209 = 47). Since 2209 is a perfect square, the solutions are rational.

Step 4: Conclude the Type of Solutions

Based on our calculations, the quadratic equation -24x² - 17x + 20 = 0 has two distinct rational solutions.

Applying the Quadratic Formula to Find the Solutions

To further solidify our understanding, let's find the actual solutions using the quadratic formula:

x = (-b ± √(b² - 4ac)) / 2a x = (17 ± √2209) / (2 * -24) x = (17 ± 47) / -48

This gives us two solutions:

x₁ = (17 + 47) / -48 = 64 / -48 = -4/3 x₂ = (17 - 47) / -48 = -30 / -48 = 5/8

As we predicted, the solutions are rational numbers, specifically -4/3 and 5/8.

Importance of Understanding the Discriminant

The discriminant is a powerful tool that offers insights into the nature of quadratic equation solutions without requiring the full application of the quadratic formula. This understanding is crucial in various mathematical and scientific fields.

Practical Applications in Mathematics and Beyond

In mathematics, the discriminant is used to analyze the behavior of quadratic functions, determine the number of real roots, and understand the graphical representation of parabolas. It’s an essential concept in algebra, calculus, and numerical analysis.

Beyond mathematics, the discriminant finds applications in physics, engineering, and computer science. For example, in physics, it can be used to analyze projectile motion, where the roots of a quadratic equation determine the time at which a projectile reaches a certain height. In engineering, it can be applied to circuit analysis and structural mechanics. In computer science, it can be used in optimization algorithms and graph theory.

Real-World Examples

Consider a scenario where an engineer is designing a bridge. The equation describing the parabolic arch of the bridge can be represented as a quadratic equation. By analyzing the discriminant, the engineer can determine whether the arch intersects the ground at two points (two real solutions), one point (one real solution), or no points (complex solutions), ensuring the structural integrity of the bridge.

In finance, quadratic equations are used to model investment returns and risk. The discriminant can help investors assess the potential for profit and loss by determining the nature of the solutions to these equations.

Conclusion

In summary, the quadratic equation -24x² - 17x + 20 = 0 has two distinct rational solutions. We arrived at this conclusion by calculating the discriminant (Δ = 2209), which is positive and a perfect square. This process demonstrates the power of the discriminant in quickly classifying the solutions of a quadratic equation.

Understanding the discriminant and its implications is essential for anyone studying algebra and its applications. It provides a valuable shortcut for analyzing quadratic equations and solving real-world problems across various disciplines. By mastering this concept, you gain a deeper understanding of the behavior of quadratic functions and their significance in the world around us.