Quadratic equations are a cornerstone of algebra, offering a window into complex mathematical concepts. These equations, typically in the form `ax² + bx + c = 0`

, are not just theoretical constructs but are pivotal in various practical applications. In this article, we delve deep into the intricacies of one such equation: `x²-11x+28=0`

, exploring its solutions, implications, and real-world applications.

Table of Contents

## Understanding the Structure of Quadratic Equations

Before tackling `x²-11x+28=0`

, it’s crucial to comprehend the standard form of a quadratic equation. It is expressed as `ax² + bx + c = 0`

, where `a`

, `b`

, and `c`

are coefficients, and `x`

represents the variable. In our equation, `a`

is 1, `b`

is -11, and `c`

is 28.

## Solving x²-11x+28=0: A Step-by-Step Approach

### Factoring the Equation

Factoring is a primary method for solving simpler quadratic equations. However, for `x²-11x+28=0`

, we need a more robust approach due to its complexity.

### The Quadratic Formula

The quadratic formula, `x = (-b ± √(b² – 4ac)) / 2a`

, is a universal solution for quadratic equations. Applying this to `x²-11x+28=0`

, we get:

**Identify Coefficients**:`a = 1`

,`b = -11`

,`c = 28`

.**Apply the Formula**:`x = (11 ± √((-11)² – 4 * 1 * 28)) / 2`

.**Simplify**:`x = (11 ± √(121 – 112)) / 2`

.**Find the Roots**:`x = (11 + 3) / 2 = 7`

and`x = (11 - 3) / 2 = 4`

.

Thus, the roots of `x²-11x+28=0`

are `x = 7`

and `x = 4`

.

## Graphical Interpretation of Quadratic Equations

```
graph LR
A[Parabola Opening Upwards] --> B((Vertex: Minimum Point))
A --> C[Roots: x=7 and x=4]
B --> D[Axis of Symmetry: x=-b/2a]
```

A quadratic equation graphically represents a parabola. For `x²-11x+28=0`

, the parabola opens upwards, with the vertex indicating the minimum point and the roots representing the points where the graph intersects the x-axis.

## Practical Applications of Quadratic Equations

Quadratic equations like `x²-11x+28=0`

are not confined to theoretical mathematics. They find extensive use in fields such as physics, engineering, and economics. For instance, in projectile motion in physics, the trajectory of an object is often modeled using a quadratic equation.

## Conclusion

The journey through the world of quadratic equations, particularly `x²-11x+28=0`

, reveals the depth and applicability of these mathematical constructs. From solving the equation using the quadratic formula to understanding its graphical representation and real-world applications, quadratic equations are an indispensable part of mathematics and its practical applications.