Solving Quadratics By Completing The Square

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Introduction

Quadratic equations are one of the most important topics in algebra. They are used in many fields of study, including physics, engineering, and finance. A quadratic equation is an equation of the form ax^2 + bx + c = 0, where a, b, and c are constants and x is the variable. One way to solve a quadratic equation is by completing the square. This method involves manipulating the equation so that it can be written in the form (x + p)^2 = q, where p and q are constants.

Completing the Square

To complete the square, we need to add and subtract a constant term to the quadratic equation. The constant term we add is half of the coefficient of x, squared. This will result in a perfect square trinomial, which can be factored into (x + p)^2 = q. The constant term we subtract is the same as the constant term we added. This will keep the equation balanced.

Example

Let's consider the quadratic equation x^2 + 6x + 5 = 0. To complete the square, we need to add and subtract (6/2)^2 = 9. The equation becomes x^2 + 6x + 9 - 9 + 5 = 0. We can now factor the perfect square trinomial x^2 + 6x + 9 as (x + 3)^2 and simplify the equation to (x + 3)^2 - 4 = 0. Solving for x, we get x = -3 ± 2.

Advantages of Completing the Square

One advantage of completing the square is that it always works. Unlike factoring or using the quadratic formula, completing the square can be used to solve any quadratic equation. Another advantage is that it helps us understand the relationship between the coefficients and roots of the quadratic equation. By manipulating the equation, we can see how the roots change as we change the coefficients.

Disadvantages of Completing the Square

One disadvantage of completing the square is that it can be time-consuming. It requires several steps and some algebraic manipulation. Another disadvantage is that it can be difficult to remember the steps. Unlike the quadratic formula, which is a simple formula that can be memorized, completing the square requires some understanding of algebraic concepts.

Applications of Completing the Square

Completing the square has many applications in mathematics and science. In physics, it is used to find the maximum height and range of projectiles. In engineering, it is used to optimize the design of structures and machines. In finance, it is used to calculate the present value of future payments.

Conclusion

Completing the square is a powerful tool for solving quadratic equations. It allows us to manipulate the equation and find its roots. While it can be time-consuming and difficult to remember, it is a valuable skill for anyone studying mathematics or science. With practice, completing the square can become second nature and help us solve problems more efficiently.