Understanding And Solving Quadratic Equations: X² + 6X + 9 = 0

How do you factor the trinomial x^2+6x+9=0? Socratic from socratic.org

Introduction

Quadratic equations are equations that contain a variable of the second degree, usually represented by x². These equations are commonly found in algebra and are used to model various real-life situations. One important aspect of quadratic equations is finding their roots or solutions, which are the values of x that make the equation true. In this article, we will explore how to solve the quadratic equation x² + 6x + 9 = 0.

The Quadratic Formula

The most common method for solving quadratic equations is using the quadratic formula:

x = (-b ± √(b² - 4ac)) / 2a

Where a, b, and c are the coefficients of the quadratic equation ax² + bx + c = 0. In our equation, x² + 6x + 9 = 0, a = 1, b = 6, and c = 9. Substituting these values into the quadratic formula, we get:

x = (-6 ± √(6² - 4(1)(9))) / 2(1)

x = (-6 ± √(36 - 36)) / 2

x = -3

Factoring the Quadratic Equation

Another method for solving quadratic equations is factoring. Factoring involves expressing the quadratic equation as a product of two linear expressions. In our equation, x² + 6x + 9 = 0, we can factor it as:

(x + 3)² = 0

This is because when we expand (x + 3)², we get x² + 6x + 9. Setting it equal to zero, we get:

(x + 3)² = 0

x + 3 = 0

x = -3

Completing the Square

Completing the square is another method for solving quadratic equations. This method involves manipulating the equation to form a perfect square trinomial, which can then be easily factored. In our equation, x² + 6x + 9 = 0, we can complete the square as follows:

x² + 6x = -9

x² + 6x + 9 = 0

(x + 3)² = 0

x + 3 = 0

x = -3

Using Graphs to Solve Quadratic Equations

Graphing the quadratic equation can also help in finding its roots. The roots of the equation are the x-intercepts of the graph. In our equation, x² + 6x + 9 = 0, we can graph it using a graphing calculator or software to find that the x-intercept is -3.

Real-World Applications of Quadratic Equations

Quadratic equations are used to model various real-life situations, such as the trajectory of a ball, the shape of a satellite dish, and the profit and loss of a business. By solving quadratic equations, we can make predictions and decisions based on these models.

Conclusion

In conclusion, the quadratic equation x² + 6x + 9 = 0 can be solved using various methods, such as the quadratic formula, factoring, completing the square, and graphing. These methods can also be applied to solve other quadratic equations and to model real-life situations. By understanding and solving quadratic equations, we can gain insights and make informed decisions.