Discriminant Of A Quadratic Equation Explained With Example

In the realm of mathematics, quadratic equations hold a significant place, appearing in various applications ranging from physics to engineering. A crucial aspect of understanding quadratic equations lies in the discriminant, a powerful tool that unveils the nature of the equation's solutions. In this article, we delve deep into the concept of the discriminant, exploring its definition, calculation, and the profound implications it holds for the solutions of quadratic equations. We will illustrate these concepts using the example equation $2x^2 - 3x + 3 = 0$, providing a step-by-step guide to computing the discriminant and interpreting its value.

What is the Discriminant?

The discriminant is a mathematical expression that provides vital information about the nature and number of solutions (also known as roots) of a quadratic equation. A quadratic equation is a polynomial equation of the second degree, generally represented in the standard form as:

$ax^2 + bx + c = 0$

where 'a', 'b', and 'c' are constants, and 'x' is the variable. The discriminant, denoted by the Greek letter delta (Δ), is calculated using the following formula:

Δ = b² - 4ac

The discriminant's value reveals whether the quadratic equation has real solutions, imaginary solutions, or a combination of both. It also indicates the number of distinct real solutions, which can be either two, one (a repeated root), or none.

Calculating the Discriminant for $2x^2 - 3x + 3 = 0$

To illustrate the practical application of the discriminant, let's consider the quadratic equation provided: $2x^2 - 3x + 3 = 0$. Our goal is to compute the discriminant (Δ) for this equation and then interpret its value to determine the nature of the solutions.

Step 1: Identify the coefficients

The first step involves identifying the coefficients 'a', 'b', and 'c' from the given quadratic equation. By comparing the equation $2x^2 - 3x + 3 = 0$ with the standard form $ax^2 + bx + c = 0$, we can readily identify the coefficients:

• a = 2
• b = -3
• c = 3

Step 2: Apply the discriminant formula

Now that we have identified the coefficients, we can proceed to calculate the discriminant (Δ) using the formula:

Δ = b² - 4ac

Substituting the values of 'a', 'b', and 'c' that we obtained in the previous step, we get:

Δ = (-3)² - 4(2)(3)

Step 3: Simplify the expression

Next, we simplify the expression to arrive at the value of the discriminant:

Δ = 9 - 24

Δ = -15

Therefore, the discriminant for the quadratic equation $2x^2 - 3x + 3 = 0$ is -15.

Interpreting the Discriminant's Value

The value of the discriminant provides valuable insights into the nature of the solutions of the quadratic equation. Here's how to interpret the discriminant:

• Δ > 0 (Positive Discriminant): If the discriminant is positive, the quadratic equation has two distinct real solutions. This means there are two different values of 'x' that satisfy the equation.
• Δ = 0 (Zero Discriminant): If the discriminant is zero, the quadratic equation has exactly one real solution, which is a repeated root. This means the quadratic equation has two identical solutions.
• Δ < 0 (Negative Discriminant): If the discriminant is negative, the quadratic equation has no real solutions. Instead, it has two complex solutions, which involve imaginary numbers.

In our example, the discriminant for the equation $2x^2 - 3x + 3 = 0$ is -15, which is negative (Δ < 0). Therefore, we can conclude that this quadratic equation has no real solutions. It has two complex solutions, which are not real numbers.

Number of Real Solutions: A Summary

Based on the discriminant's value, we can summarize the number of real solutions as follows:

• If Δ > 0: Two distinct real solutions
• If Δ = 0: One real solution (repeated root)
• If Δ < 0: No real solutions (two complex solutions)

For the equation $2x^2 - 3x + 3 = 0$, the discriminant is -15, which is negative. Therefore, the number of real solutions is zero.

Visualizing the Solutions

To further enhance our understanding, let's visualize the solutions of a quadratic equation graphically. The solutions of a quadratic equation correspond to the points where the parabola (the graph of the quadratic equation) intersects the x-axis.

• Two Distinct Real Solutions (Δ > 0): The parabola intersects the x-axis at two distinct points.
• One Real Solution (Repeated Root) (Δ = 0): The parabola touches the x-axis at one point (the vertex of the parabola lies on the x-axis).
• No Real Solutions (Δ < 0): The parabola does not intersect the x-axis.

For the equation $2x^2 - 3x + 3 = 0$, since the discriminant is negative, the parabola does not intersect the x-axis, confirming that there are no real solutions.

The Discriminant and the Quadratic Formula

The discriminant plays a pivotal role in the quadratic formula, which is used to find the solutions of any quadratic equation. The quadratic formula is given by:

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

Notice that the expression under the square root in the quadratic formula is the discriminant (b² - 4ac). This highlights the discriminant's significance in determining the nature of the solutions.

• If Δ > 0, the square root of the discriminant is a real number, leading to two distinct real solutions.
• If Δ = 0, the square root of the discriminant is zero, resulting in one real solution (repeated root).
• If Δ < 0, the square root of the discriminant is an imaginary number, leading to two complex solutions.

Real-World Applications of the Discriminant

The discriminant is not merely an abstract mathematical concept; it has practical applications in various fields, including:

• Physics: In physics, quadratic equations are used to model projectile motion, and the discriminant can help determine whether a projectile will reach a certain height or not.
• Engineering: Engineers use quadratic equations in designing structures and systems, and the discriminant can help assess the stability and feasibility of a design.
• Economics: Quadratic equations are used in economic models, and the discriminant can provide insights into the equilibrium points and stability of economic systems.
• Computer Graphics: In computer graphics, quadratic equations are used for curve and surface modeling, and the discriminant can help determine the intersection points of curves and surfaces.

Common Mistakes to Avoid

When working with the discriminant, it's crucial to avoid common mistakes to ensure accurate results. Some common errors include:

• Incorrectly Identifying Coefficients: Ensure that you correctly identify the values of 'a', 'b', and 'c' from the quadratic equation. Pay close attention to the signs (+ or -) of the coefficients.
• Misapplying the Discriminant Formula: Double-check that you are using the correct discriminant formula (Δ = b² - 4ac) and that you are substituting the values correctly.
• Misinterpreting the Discriminant's Value: Understand the meaning of the discriminant's value (positive, zero, or negative) and how it relates to the number and nature of the solutions.
• Arithmetic Errors: Be careful with arithmetic operations, especially when dealing with squares, multiplications, and subtractions. A small arithmetic error can lead to an incorrect discriminant value and, consequently, an incorrect interpretation of the solutions.

Conclusion

The discriminant is a powerful tool for analyzing quadratic equations, providing valuable information about the nature and number of solutions. By calculating the discriminant (Δ = b² - 4ac), we can determine whether a quadratic equation has two distinct real solutions, one real solution (repeated root), or no real solutions (two complex solutions). Understanding the discriminant and its implications is crucial for solving quadratic equations effectively and for applying them in various real-world contexts. In this article, we have explored the concept of the discriminant in detail, using the example equation $2x^2 - 3x + 3 = 0$ to illustrate the calculation and interpretation process. We hope this comprehensive guide has provided you with a deeper understanding of the discriminant and its significance in the world of mathematics and beyond.

By mastering the discriminant, you gain a valuable tool for solving quadratic equations and understanding their applications in various fields. Remember to carefully identify the coefficients, apply the discriminant formula correctly, and interpret the discriminant's value accurately to unlock the secrets hidden within quadratic equations.