Completing the Square: A Guide to Transforming Quadratic Expressions into Perfect Squares

Understanding how to complete the square is an essential skill in algebra. This method is not only a foundational concept in mathematics but also plays a critical role in solving a variety of problems, including optimization and finding the vertex of a parabola.

Introduction to Completing the Square

Completing the square is a technique used to transform a quadratic expression into a perfect square trinomial. A perfect square trinomial is a quadratic expression that can be expressed as the square of a binomial.

Step-by-Step Guide to Completing the Square

Step 1: Ensure the Leading Coefficient is Unity

For simplicity, start by ensuring that the coefficient of the x^2 term is 1. If it's not, factor out the leading coefficient from the x^2 and x terms.

For example, consider the expression 2x^2 - 17. The leading coefficient is 2, so factor out 2:

2x^2 - 17 2(x^2 - 8.5)

Step 2: Complete the Square for the Inner Expression

To complete the square, you need to find a constant to add and subtract within the inner expression, making it a perfect square trinomial.

Recall that a perfect square trinomial is of the form:

(ax - r)^2 a x^2 - 2arx r^2

Here, we need to find r such that:

2x^2 - 17 (constant) 2(x^2 - 4.25)

We can use the formula:

constant (r)^2 / (a) (5/2)^2 / 2 25/4 * 1/2 25/8

Therefore, we add and subtract 25/8 from the expression:

2x^2 - 17 2[x^2 - 8.5 25/8 - 25/8] 2[x^2 - 4.25 3.125 - 25/8]

The expression inside the brackets can be written as:

2(x^2 - 4.25 3.125 - 25/8) 2[(x - 5/2)^2 - 25/8]

Step 3: Simplify and Rearrange

Simplify the expression to get a perfect square trinomial:

2x^2 - 17 2[(x - 5/2)^2 - 25/8]

Therefore, the expression becomes:

2x^2 - 17 - 25/8 2(x - 5/2)^2 - 25/4

Adding 25/4 to the expression to make it a perfect square trinomial:

2x^2 - 17 25/4 2[(x - 5/2)^2]

The final form is:

2x^2 - 17 25/2 2(x - 5/2)^2

Key Points and Examples

Key Points

Factor out the leading coefficient: This step simplifies the process of completing the square. Find the perfect square trinomial: Use the formula (ax - r)^2 a x^2 - 2arx r^2 to identify the constant. Adjust the expression: Add and subtract the constant identified in step 2 to the expression. Simplify and rearrange: Simplify the expression and rearrange it to express it as a perfect square trinomial.

Example

Consider the expression 2x^2 - 17 2(x^2 - 8.5). To make it a perfect square, we follow the steps:

Identify the constant to be added: 25/8. Factor out the constant: 2[(x^2 - 4.25 3.125 - 25/8)]. Complete the square: 2[(x - 5/2)^2 - 25/8]. Simplify: 2x^2 - 17 2[(x - 5/2)^2 - 25/8]. Final expression: 2(x - 5/2)^2 when 25/4 is added.

Practice and Application

To further practice and apply this method, consider completing the square with different quadratic expressions, such as:

Example 1: 4x^2 - 24x - 20 Factor out the leading coefficient: 4(x^2 - 6x - 5). Complete the square: 4[x^2 - 6x 9 - 14] 4[(x - 3)^2 - 14]. Add and subtract 9: 4(x^2 - 6x 9) - 56 36 4(x - 3)^2 - 20.

Example 2: 3x^2 12x - 15 Factor out the leading coefficient: 3(x^2 4x - 5). Complete the square: 3[x^2 4x 4 - 9] 3[(x 2)^2 - 9]. Add and subtract 4: 3(x^2 4x 4) - 27 12 3(x 2)^2 - 15.

These examples illustrate the step-by-step process of completing the square and transforming quadratic expressions into perfect squares.

Conclusion

The method of completing the square is a powerful mathematical tool with numerous applications, from solving equations to analyzing functions. By mastering this technique, you can solve a wide range of problems with confidence and accuracy.