## Introduction to Quadratic Equations

Quadratic equations appear in numerous aspects of both mathematics and real-world problems. A quadratic equation is any equation that can be written in the form:

where , , and are constants, and . Some typical real-life applications of quadratic equations include projectile motion, optimizing areas, and economics.

In this article, we will explore one particular method of solving quadratic equations: completing the square. This method not only provides a systematic approach to solving quadratics but also lays the foundation for understanding the quadratic formula.

## Fundamentals of Completing the Square

Completing the square involves rearranging a quadratic equation so that one side of the equation becomes a perfect square trinomial. A perfect square trinomial is an expression that can be factored into a binomial squared. For the equation

we will follow these steps:

**Move the constant term to the other side:**Rewrite the equation as .**Divide by :**Simplify to get .**Complete the square:**Add and subtract inside the equation.**Factor and solve:**Convert the left side to a binomial square, then solve the resulting equation.

## Example: Solving a Quadratic Equation

Let’s work through a concrete example to illustrate these steps. Consider the quadratic equation:

- Move the constant term to the other side:
- Divide by the coefficient of :
- Complete the square. Add and subtract :
- Factor and solve. The left side is now a perfect square trinomial:
Taking the square root of both sides, we get:

Thus,

Therefore, the solutions to the equation are and .

## Applications in Real Life

Quadratic equations appear in various real-world situations, and completing the square can help solve such problems. Consider a scenario where an architect is designing a rectangular garden and needs to optimize the area under specific length constraints.

If the length of the garden is meters and the width is meters, the area can be represented by the quadratic equation:

If the total area is constrained to be 50 square meters, we get:

By completing the square here, the architect can find the dimensions that satisfy the area requirements.

## Summary and Conclusion

Completing the square is a powerful method for solving quadratic equations. It enhances understanding of quadratic properties and prepares students for the quadratic formula.

- It provides a systematic method for solving any quadratic equation.
- It is foundational for deriving the quadratic formula.
- It has practical applications in various fields such as physics, engineering, and economics.

We hope this article has provided clear insights into solving quadratic equations by completing the square, empowering you to tackle both academic and real-world challenges.