Completing the Square: Turning x2 - 8x into a Perfect Square Trinomial

One of the key concepts in algebra is mastering the technique of completing the square. This involves manipulating an expression to create a perfect square trinomial, which is of the form a^2 - 2ab b^2. This tutorial will guide you through the process of turning x^2 - 8x into a perfect square trinomial.

Understanding a Perfect Square Trinomial

A perfect square trinomial is a quadratic expression that can be factored into the square of a binomial. For example, a^2 - 2ab b^2 can be factored as (a - b)^2. This is a crucial concept in solving quadratic equations and factoring expressions.

Steps to Complete the Square

Identify the coefficient of x.

Take half of this coefficient and square it.

Add this value to the expression and also subtract it to maintain equality.

Factor the resulting expression to show it as a perfect square trinomial.

Applying the Steps to x2 - 8x

In the expression x^2 - 8x, the coefficient of x is -8.

Half of -8 is -4, and squaring this gives 16.

To complete the square, we add 16 to the expression and also add 16 on the other side of the equation (though not necessary for an expression).

The expression now becomes x^2 - 8x 16.

This can be factored as (x - 4)^2.

Conceptual Insight

The reason we add 16 is to create a perfect square trinomial. When we have the form a^2 - 2ab b^2, we know that a x and 2ab 8x. Solving for b gives b -4. Consequently, b^2 16.

Conclusion

Adding 16 to the expression x^2 - 8x makes it a perfect square trinomial, specifically (x - 4)^2. This process can be applied to any quadratic expression to transform it into a perfect square trinomial.

Additional Example

Let's consider another example, x^2 - 1 - 319. To make it a perfect square trinomial, we follow the same steps:

Identify the coefficient of x, which is -10.

Half of -10 is -5, and squaring this gives 25.

Add 25 to the expression and also add 25 on the other side of the equation.

The expression becomes x^2 - 1 25 - 319 25 (for the equation) or simply x^2 - 1 25 for the expression itself.

This can be factored as (x - 5)^2.

The solution depends on the context, but the method remains consistent.