(2x+1)^2+(3x-1)^2+2(2x+1)(3x-1)

2 min read Jun 16, 2024

Simplifying the Expression (2x + 1)² + (3x - 1)² + 2(2x + 1)(3x - 1)

This expression might look daunting at first, but it can be simplified using the algebraic identity for the square of a binomial and the distributive property. Let's break down the process step-by-step:

Understanding the Identity

The key to simplifying this expression lies in recognizing the pattern of the first three terms:

(a + b)² = a² + 2ab + b²

This identity states that squaring a binomial (a + b) results in the sum of the square of the first term (a²), twice the product of the first and second term (2ab), and the square of the second term (b²).

Applying the Identity

Let's apply this identity to our expression:

(2x + 1)²: Using the identity, we get (2x)² + 2(2x)(1) + (1)² = 4x² + 4x + 1
(3x - 1)²: Applying the identity, we get (3x)² + 2(3x)(-1) + (-1)² = 9x² - 6x + 1
2(2x + 1)(3x - 1): This term is already in a form that allows us to directly apply the distributive property. Let's expand it: 2[(2x)(3x) + (2x)(-1) + (1)(3x) + (1)(-1)] = 2(6x² - 2x + 3x - 1) = 12x² + 2x - 2

Combining the Terms

Now, let's add all the simplified terms together:

(4x² + 4x + 1) + (9x² - 6x + 1) + (12x² + 2x - 2) = 25x² + 6x

The Simplified Expression

Therefore, the simplified form of the expression (2x + 1)² + (3x - 1)² + 2(2x + 1)(3x - 1) is 25x² + 6x.