X 2 6x 9 0 Graph

Unveiling the Secrets of the x² - 6x + 9 = 0 Graph: A Comprehensive Guide

The quadratic equation x² - 6x + 9 = 0 might seem simple at first glance, but it holds a wealth of mathematical concepts and visual representations that are crucial for understanding more complex algebraic and graphical relationships. This article delves deep into the intricacies of this seemingly straightforward equation, exploring its roots, vertex, axis of symmetry, and graphical interpretation. We will also explore how this specific equation relates to broader quadratic concepts and its applications in various fields.

Understanding the Quadratic Equation

Before we dive into the specifics of x² - 6x + 9 = 0, let's establish a foundational understanding of quadratic equations. A quadratic equation is a polynomial equation of the second degree, meaning the highest power of the variable (x in this case) is 2. The general form of a quadratic equation is ax² + bx + c = 0, where a, b, and c are constants, and a ≠ 0.

In our specific equation, x² - 6x + 9 = 0, we have a = 1, b = -6, and c = 9. The values of a, b, and c dictate the shape and position of the parabola represented by the equation. The value of 'a' determines whether the parabola opens upwards (a > 0) or downwards (a < 0). In our case, since a = 1 (positive), the parabola opens upwards.

Solving the Quadratic Equation: Finding the Roots

The roots (or solutions) of a quadratic equation are the values of x that satisfy the equation, making it equal to zero. There are several methods to solve quadratic equations, including factoring, the quadratic formula, and completing the square.

Factoring: This method involves rewriting the equation as a product of two linear expressions. Our equation, x² - 6x + 9 = 0, is a perfect square trinomial, which means it can be factored easily:

(x - 3)(x - 3) = 0 or (x - 3)² = 0

This reveals that the equation has a single, repeated root: x = 3.

Quadratic Formula: The quadratic formula provides a general solution for any quadratic equation:

x = [-b ± √(b² - 4ac)] / 2a

Substituting the values from our equation (a = 1, b = -6, c = 9), we get:

x = [6 ± √((-6)² - 4 * 1 * 9)] / (2 * 1) = [6 ± √(36 - 36)] / 2 = 6 / 2 = 3

Again, we find that the only root is x = 3. This indicates that the parabola touches the x-axis at only one point.

Graphical Representation: The Parabola

The graph of a quadratic equation is always a parabola. The parabola's characteristics are directly related to the coefficients of the quadratic equation.

Vertex: The vertex is the turning point of the parabola. For a parabola in the form ax² + bx + c, the x-coordinate of the vertex is given by:

x = -b / 2a

In our equation, this gives:

x = -(-6) / (2 * 1) = 3

The y-coordinate of the vertex is found by substituting the x-coordinate back into the equation:

y = (3)² - 6(3) + 9 = 0

Therefore, the vertex of the parabola is (3, 0). This confirms that the parabola touches the x-axis at x = 3.

Axis of Symmetry: The axis of symmetry is a vertical line that passes through the vertex, dividing the parabola into two symmetrical halves. Its equation is given by:

x = -b / 2a

For our equation, the axis of symmetry is x = 3.

y-intercept: The y-intercept is the point where the parabola intersects the y-axis (where x = 0). Substituting x = 0 into the equation:

y = (0)² - 6(0) + 9 = 9

The y-intercept is (0, 9).

Plotting the Graph: Using the vertex (3, 0), the axis of symmetry (x = 3), the y-intercept (0, 9), and the knowledge that the parabola opens upwards, we can accurately plot the graph of x² - 6x + 9 = 0. The parabola is a U-shaped curve that touches the x-axis at the point (3, 0).

The Significance of a Single Root

The fact that x² - 6x + 9 = 0 has only one repeated root (x = 3) has significant implications. This indicates that the parabola is tangent to the x-axis at the point (3,0). It doesn't intersect the x-axis at two distinct points. This is a special case of quadratic equations where the discriminant (b² - 4ac) equals zero. The discriminant is a crucial part of the quadratic formula, determining the nature of the roots.

• Discriminant > 0: Two distinct real roots (parabola intersects x-axis at two points)
• Discriminant = 0: One repeated real root (parabola is tangent to the x-axis)
• Discriminant < 0: Two distinct complex roots (parabola does not intersect x-axis)

In our equation, the discriminant is (-6)² - 4(1)(9) = 0, hence the single repeated root.

Applications and Extensions

The understanding of quadratic equations and their graphical representations extends far beyond simple algebraic solutions. They have numerous applications in various fields, including:

• Physics: Projectile motion, where the trajectory of a thrown object can be modeled using a quadratic equation.
• Engineering: Designing parabolic reflectors for antennas and telescopes.
• Economics: Modeling supply and demand curves, and determining optimal production levels.
• Computer Graphics: Creating curved shapes and animations.

Furthermore, the concept of a perfect square trinomial, as seen in our factored equation (x - 3)², is crucial in various algebraic manipulations and simplifications. Understanding perfect squares simplifies the process of solving more complex equations and inequalities.

Conclusion: A Deeper Appreciation of x² - 6x + 9 = 0

The seemingly simple equation x² - 6x + 9 = 0 provides a rich foundation for understanding quadratic equations, their graphical representations, and their applications. By exploring its roots, vertex, axis of symmetry, and the significance of its single repeated root, we gain a deeper appreciation for the interplay between algebra and geometry. This understanding is crucial for tackling more complex mathematical problems and for appreciating the power of quadratic equations in various real-world applications. The analysis of this specific equation serves as a valuable stepping stone towards mastering more advanced concepts in mathematics and related fields. The exploration of its graphical representation reinforces the visual interpretation of algebraic concepts, making the learning process more intuitive and engaging. Understanding this simple quadratic equation provides a solid foundation for more advanced mathematical pursuits.