Discriminant Of Quadratic Equation $x^2+11x+121=x+96$

In the realm of mathematics, quadratic equations hold a position of immense importance. These equations, characterized by the presence of a squared variable, appear in a myriad of applications, ranging from physics and engineering to economics and computer science. Understanding the nature of their solutions is crucial for solving a wide array of problems.

Decoding the Quadratic Equation

A quadratic equation is a polynomial equation of the second degree, generally expressed in the form:

$ax^2 + bx + c = 0$

where a, b, and c are constants, and a is not equal to zero. The solutions to this equation, also known as the roots or zeros, represent the values of x that satisfy the equation. These roots can be real or complex numbers, and their nature is determined by a critical component called the discriminant.

The discriminant, often denoted by the Greek letter Delta (Δ), is a mathematical expression that provides valuable insights into the nature of the roots of a quadratic equation. It is calculated using the coefficients a, b, and c from the quadratic equation:

Δ = $b^2 - 4ac$

The discriminant's value reveals whether the quadratic equation has two distinct real roots, one real root (a repeated root), or two complex roots. Let's delve deeper into the significance of the discriminant's value.

The Discriminant's Tale of Roots

The discriminant acts as a guide, revealing the secrets of a quadratic equation's solutions. Its value dictates the type and number of roots the equation possesses:

• Δ > 0 (Positive Discriminant): When the discriminant is positive, the quadratic equation boasts two distinct real roots. This means there are two different values of x that satisfy the equation. These roots represent the points where the parabola represented by the quadratic equation intersects the x-axis.
• Δ = 0 (Zero Discriminant): A zero discriminant signifies that the quadratic equation has exactly one real root, often referred to as a repeated root or a double root. In this scenario, the parabola touches the x-axis at a single point, indicating that the two roots have coalesced into one.
• Δ < 0 (Negative Discriminant): When the discriminant is negative, the quadratic equation ventures into the realm of complex numbers. It possesses two complex roots, which are complex conjugates of each other. These roots do not appear on the real number line, and the parabola does not intersect the x-axis.

Calculating the Discriminant A Step-by-Step Guide

To calculate the discriminant, follow these straightforward steps:

1. Identify the coefficients: Begin by identifying the coefficients a, b, and c from the quadratic equation in the standard form $ax^2 + bx + c = 0$.
2. Apply the formula: Substitute the values of a, b, and c into the discriminant formula: Δ = $b^2 - 4ac$.
3. Compute the result: Perform the arithmetic operations to calculate the value of the discriminant.
4. Interpret the value: Analyze the discriminant's value to determine the nature of the roots.

A Practical Example: Finding the Discriminant

Let's consider the quadratic equation provided: $x^2 + 11x + 121 = x + 96$. To find the discriminant, we first need to rewrite the equation in the standard form:

$x^2 + 11x + 121 - x - 96 = 0$

Simplifying the equation, we get:

$x^2 + 10x + 25 = 0$

Now, we can identify the coefficients:

• a = 1
• b = 10
• c = 25

Next, we apply the discriminant formula:

Δ = $b^2 - 4ac$ = $(10)^2 - 4(1)(25)$ = $100 - 100$ = 0

Therefore, the discriminant of the quadratic equation $x^2 + 11x + 121 = x + 96$ is 0. This indicates that the equation has one real root.

Solving the Quadratic Equation

Having calculated the discriminant, we now know that the quadratic equation $x^2 + 10x + 25 = 0$ has one real root. To find this root, we can employ the quadratic formula:

x = rac{-b ± extstyle igtriangleup rac{ extstyle extstyle igtriangleup}{ extstyle 2a}}{2a}

Substituting the values of a, b, and the discriminant (Δ = 0), we get:

x = rac{-10 ± extstyle igtriangleup rac{ extstyle extstyle igtriangleup}{ extstyle 2(1)}}{2(1)} = rac{-10 ± 0}{2} = -5

Thus, the quadratic equation has one real root, x = -5.

The Discriminant in Action Real-World Applications

The discriminant is not merely a theoretical concept; it finds practical applications in various fields. For instance, in physics, it helps determine the nature of projectile motion. In engineering, it aids in analyzing the stability of structures. In computer graphics, it plays a role in collision detection algorithms. The discriminant's ability to predict the nature of solutions makes it a valuable tool in problem-solving across diverse domains.

Projectile Motion

In physics, the trajectory of a projectile, such as a ball thrown into the air, can be modeled using a quadratic equation. The discriminant can then be used to determine whether the projectile will hit a target, reach a certain height, or follow a specific path. A positive discriminant indicates that the projectile will intersect the target at two points, a zero discriminant indicates that it will intersect at one point (a perfect hit), and a negative discriminant indicates that it will miss the target.

Structural Stability

In engineering, the stability of structures, such as bridges and buildings, can be analyzed using quadratic equations. The discriminant helps determine whether a structure will withstand certain loads or forces. A positive discriminant might indicate potential instability, while a zero or negative discriminant suggests greater stability.

Collision Detection

In computer graphics and game development, collision detection is a crucial aspect. Determining whether two objects will collide often involves solving quadratic equations. The discriminant can then be used to efficiently determine whether a collision will occur, allowing for realistic interactions between objects in a virtual environment.

Mastering the Discriminant A Key to Quadratic Equations

The discriminant is a fundamental concept in the study of quadratic equations. Its ability to reveal the nature of the roots empowers us to solve a wide range of problems in mathematics and other disciplines. By understanding the discriminant's significance and mastering its calculation, we unlock a deeper understanding of quadratic equations and their applications.

In conclusion, the discriminant is an indispensable tool in the realm of quadratic equations. Its value provides critical information about the nature of the roots, guiding us to solve equations and tackle real-world problems. By grasping the concept of the discriminant, we gain a powerful advantage in our mathematical journey.

Further Exploration

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

• The quadratic formula and its derivation
• Graphing quadratic equations and the relationship to the discriminant
• Applications of quadratic equations in various fields
• Solving quadratic inequalities

By delving deeper into these areas, you will solidify your grasp of quadratic equations and their significance in mathematics and beyond.

Conclusion: The Power of the Discriminant

The discriminant stands as a testament to the elegance and power of mathematical concepts. Its ability to unveil the nature of quadratic equation solutions makes it an invaluable tool for mathematicians, scientists, engineers, and anyone who seeks to understand the world through the lens of mathematics. Master the discriminant, and you master a key to unlocking the secrets of quadratic equations.