What Is The Discriminant Of 9x2 2 10x

What is the Discriminant of 9x² + 10x?

The discriminant is a powerful tool in algebra, particularly when dealing with quadratic equations. It provides crucial information about the nature of the roots (solutions) of a quadratic equation without actually solving the equation. Understanding the discriminant allows us to determine whether a quadratic equation has real, distinct roots; real, repeated roots; or complex roots. This article will delve into the concept of the discriminant, focusing specifically on the quadratic equation 9x² + 10x. We'll explore its calculation, interpretation, and broader implications within the context of quadratic equations.

Understanding Quadratic Equations and Their Roots

Before diving into the discriminant, let's refresh our understanding of quadratic equations. A quadratic equation is an equation of the form:

ax² + bx + c = 0

where 'a', 'b', and 'c' are constants, and 'a' is not equal to zero. The solutions, or roots, of this equation represent the x-values where the quadratic function intersects the x-axis. These roots can be found using the quadratic formula:

x = [-b ± √(b² - 4ac)] / 2a

The expression within the square root, b² - 4ac, is the discriminant.

Calculating the Discriminant for 9x² + 10x

Our specific quadratic equation is 9x² + 10x. Comparing this to the standard form ax² + bx + c = 0, we can identify the coefficients:

• a = 9
• b = 10
• c = 0 (Note that the constant term is missing, implying c = 0)

Now, let's substitute these values into the discriminant formula:

Discriminant (D) = b² - 4ac = (10)² - 4 * (9) * (0) = 100 - 0 = 100

Therefore, the discriminant of the quadratic equation 9x² + 10x is 100.

Interpreting the Discriminant: What Does 100 Tell Us?

The value of the discriminant directly informs us about the nature of the roots of the quadratic equation:

• D > 0 (Discriminant is positive): The quadratic equation has two distinct real roots. This means the parabola intersects the x-axis at two different points. In our case, since D = 100 (a positive value), the equation 9x² + 10x has two distinct real roots.

• D = 0 (Discriminant is zero): The quadratic equation has one real root (or two equal real roots). The parabola touches the x-axis at exactly one point.

• D < 0 (Discriminant is negative): The quadratic equation has two complex roots (conjugate pairs). The parabola does not intersect the x-axis.

Since the discriminant of 9x² + 10x is 100 (positive), we know it has two distinct real roots.

Finding the Roots of 9x² + 10x

Now that we know the nature of the roots, let's actually find them using the quadratic formula:

x = [-b ± √(b² - 4ac)] / 2a

Substituting the values a = 9, b = 10, and c = 0:

x = [-10 ± √(100)] / (2 * 9)

x = [-10 ± 10] / 18

This gives us two distinct solutions:

• x₁ = (-10 + 10) / 18 = 0
• x₂ = (-10 - 10) / 18 = -20/18 = -10/9

Therefore, the roots of the quadratic equation 9x² + 10x are 0 and -10/9. These values confirm our earlier deduction based on the positive discriminant.

Graphical Representation

Visualizing the quadratic equation 9x² + 10x graphically helps solidify our understanding. The graph of this equation is a parabola that opens upwards (since a = 9 > 0). The x-intercepts represent the roots of the equation. Because the discriminant is positive, the parabola intersects the x-axis at two distinct points, corresponding to the roots 0 and -10/9.

Applications of the Discriminant

The discriminant's application extends far beyond simply determining the nature of quadratic roots. It plays a vital role in various mathematical and scientific fields:

• Geometry: Determining the intersection points of lines and curves.
• Physics: Analyzing projectile motion and other physical phenomena described by quadratic equations.
• Engineering: Solving problems related to structural design, circuit analysis, and control systems.
• Computer graphics: Used in algorithms for rendering curves and surfaces.

Advanced Concepts and Extensions

The concept of the discriminant can be extended to higher-order polynomial equations. While the formula itself changes for higher-degree polynomials, the fundamental idea of determining the nature of roots based on a specific expression remains. For instance, the discriminant of a cubic equation provides information about the number of real and complex roots.

Conclusion

The discriminant is an indispensable tool for analyzing quadratic equations. By simply calculating b² - 4ac, we gain invaluable insight into the nature and number of solutions without the need for extensive calculations. Understanding the discriminant is crucial for anyone working with quadratic equations, and its applications extend far beyond the realm of pure mathematics. The specific example of 9x² + 10x, with its discriminant of 100, clearly demonstrates how a positive discriminant leads to two distinct real roots, a fact confirmed by solving the equation and visualizing its graphical representation. This fundamental understanding forms a cornerstone of algebraic problem-solving and has wide-ranging implications across various fields of study. Mastering the discriminant empowers you to solve quadratic equations efficiently and interpret their solutions accurately.