(3x-2y)^2+4(3x-2y)+4

2 min read Jun 16, 2024

Factoring the Expression (3x-2y)^2 + 4(3x-2y) + 4

This expression can be factored using the concept of perfect square trinomials.

Understanding Perfect Square Trinomials

A perfect square trinomial is a trinomial that results from squaring a binomial. The general form is:

(a + b)^2 = a^2 + 2ab + b^2

(a - b)^2 = a^2 - 2ab + b^2

Factoring the Expression

Identify the pattern: Observe that the expression (3x-2y)^2 + 4(3x-2y) + 4 resembles the pattern of a perfect square trinomial. We have a squared term, a linear term, and a constant term.
Factor the squared term: (3x-2y)^2 is already factored.
Check for the middle term: The middle term, 4(3x-2y), is twice the product of the square roots of the first and last terms: 2 * (3x-2y) * 2.
Factor the expression: Now we can rewrite the expression as a perfect square:

(3x-2y)^2 + 4(3x-2y) + 4 = (3x-2y)^2 + 2(3x-2y)(2) + 2^2

Apply the perfect square trinomial formula: This fits the pattern of (a + b)^2, where a = (3x-2y) and b = 2. Therefore, we can factor it as:

(3x-2y)^2 + 2(3x-2y)(2) + 2^2 = (3x - 2y + 2)^2

Conclusion

The factored form of the expression (3x-2y)^2 + 4(3x-2y) + 4 is (3x - 2y + 2)^2.