Completing the Square: A Step-by-Step Guide

Completing the square is a fundamental algebraic technique used to solve quadratic equations and rewrite them in vertex form. It involves manipulating the equation to create a perfect square trinomial, which can then be factored easily. This method is particularly useful when the quadratic equation cannot be factored directly.

Understanding the Concept

A perfect square trinomial is a trinomial that can be factored as the square of a binomial. For example, x² + 6x + 9 is a perfect square trinomial because it can be factored as (x + 3)². The key to completing the square is to recognize that half of the coefficient of the x term, squared, will always be the constant term needed to create a perfect square trinomial.

Steps to Complete the Square

Here's a step-by-step guide on how to complete the square:

1. Move the constant term to the right side of the equation. For example, if your equation is x² + 6x - 7 = 0, move the -7 to the right side: x² + 6x = 7.
2. Take half of the coefficient of the x term, square it, and add it to both sides of the equation. In our example, the coefficient of the x term is 6. Half of 6 is 3, and 3² is 9. So, we add 9 to both sides: x² + 6x + 9 = 7 + 9.
3. Factor the left side of the equation as a perfect square trinomial. In our example, the left side becomes (x + 3)². The right side simplifies to 16.
4. Take the square root of both sides of the equation. Remember to include both positive and negative square roots. This gives us x + 3 = ±4.
5. Solve for x. Subtract 3 from both sides to get x = -3 ± 4.

Example

Let's solve the quadratic equation x² - 8x + 12 = 0 by completing the square:

1. Move the constant term: x² - 8x = -12
2. Take half of the coefficient of the x term (-8), square it (16), and add it to both sides: x² - 8x + 16 = -12 + 16
3. Factor the left side: (x - 4)² = 4
4. Take the square root of both sides: x - 4 = ±2
5. Solve for x: x = 4 ± 2

Therefore, the solutions to the equation x² - 8x + 12 = 0 are x = 6 and x = 2.

Practice Questions

Try completing the square on these equations:

1. x² + 4x - 5 = 0
2. x² - 10x + 21 = 0
3. 2x² + 12x + 10 = 0

Conclusion

Completing the square is a powerful technique for solving quadratic equations and rewriting them in vertex form. By understanding the steps involved and practicing, you can master this important algebraic concept. Remember, practice makes perfect!