(9x-1)^2+(1-5x)^2+2(9x-1)(1-5x)

Jasonbradley

2 min read

·

Jun 16, 2024

31

Share

Factoring a Perfect Square Trinomial

The expression (9x - 1)² + (1 - 5x)² + 2(9x - 1)(1 - 5x) appears to be complex, but it can be simplified by recognizing its pattern. This expression is a perfect square trinomial, which is a special type of trinomial that results from squaring a binomial.

Here's how to identify and factor this expression:

1. Recognizing the Pattern:

The pattern of a perfect square trinomial is:

(a + b)² = a² + 2ab + b² or (a - b)² = a² - 2ab + b²

In our expression, we have:

• (9x - 1)²: This is the square of the first term (a²).
• (1 - 5x)²: This is the square of the second term (b²).
• 2(9x - 1)(1 - 5x): This is twice the product of the first and second terms (2ab).

2. Applying the Pattern:

Let's substitute a = (9x - 1) and b = (1 - 5x) into the perfect square trinomial pattern:

(a + b)² = a² + 2ab + b²

(9x - 1 + 1 - 5x)² = (9x - 1)² + 2(9x - 1)(1 - 5x) + (1 - 5x)²

3. Simplifying the Expression:

Now, we can simplify the expression:

(4x)² = (9x - 1)² + 2(9x - 1)(1 - 5x) + (1 - 5x)²

**(4x)² = (9x - 1)² + (1 - 5x)² + 2(9x - 1)(1 - 5x)

Therefore, the factored form of the expression (9x - 1)² + (1 - 5x)² + 2(9x - 1)(1 - 5x) is (4x)².

Important Note: Recognizing patterns like this can significantly simplify algebraic expressions and make solving equations easier.