How to Factor a Perfect Square Trinomial
Perfect square trinomials are a special type of quadratic expression that can be factored into two identical binomials. This article will guide you through the process of factoring a perfect square trinomial step by step.
Understanding the Structure
A perfect square trinomial always follows the pattern of (a + b)^2 = a^2 + 2ab + b^2, where ‘a’ and ‘b’ are real numbers. The first and last terms are perfect squares, and the middle term is twice the product of the square roots of the first and last terms.
Identifying the Perfect Square Trinomial
To factor a perfect square trinomial, you first need to identify whether the given expression fits the pattern. Check if the first and last terms are perfect squares and if the middle term is twice the product of the square roots of the first and last terms.
Example 1: Factoring x^2 + 6x + 9
In this example, the first term (x^2) is a perfect square (the square of x), and the last term (9) is also a perfect square (the square of 3). The middle term (6x) is twice the product of the square roots of the first and last terms (2 x 3). Therefore, this is a perfect square trinomial.
Factoring the Perfect Square Trinomial
Now that we have identified the perfect square trinomial, we can factor it by using the following formula:
(a + b)^2 = a^2 + 2ab + b^2
In our example, a = x and b = 3. Substituting these values into the formula, we get:
(x + 3)^2 = x^2 + 2(x)(3) + 3^2
(x + 3)^2 = x^2 + 6x + 9
So, the factored form of the perfect square trinomial x^2 + 6x + 9 is (x + 3)^2.
Example 2: Factoring y^2 – 4y + 4
In this example, the first term (y^2) is a perfect square (the square of y), and the last term (4) is also a perfect square (the square of 2). The middle term (-4y) is twice the product of the square roots of the first and last terms (-2 y 2). Therefore, this is a perfect square trinomial.
To factor it, we use the formula:
(a – b)^2 = a^2 – 2ab + b^2
In our example, a = y and b = 2. Substituting these values into the formula, we get:
(y – 2)^2 = y^2 – 2(y)(2) + 2^2
(y – 2)^2 = y^2 – 4y + 4
So, the factored form of the perfect square trinomial y^2 – 4y + 4 is (y – 2)^2.
Conclusion
Factoring a perfect square trinomial is a straightforward process as long as you can identify the pattern and apply the appropriate formula. By following the steps outlined in this article, you will be able to factor any perfect square trinomial with ease.
