(x-1)^2-2(x-1)(x+2)+(x+2)^2+5(2x-3)

Jasonbradley

2 min read

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Jun 17, 2024

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Simplifying the Expression: (x-1)^2-2(x-1)(x+2)+(x+2)^2+5(2x-3)

This expression looks intimidating, but with the right steps, it can be simplified. Let's break it down:

Recognizing a Pattern:

The first three terms of the expression resemble the expansion of a squared binomial. Specifically, it looks like the expansion of **(a - b)**², where:

• a = (x - 1)
• b = (x + 2)

Let's verify this by expanding (a - b)²:

(a - b)² = a² - 2ab + b²

Substituting our values for 'a' and 'b':

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

This confirms our initial observation!

Simplifying the Expression:

Now, let's simplify the expression by applying the expansion we derived:

(x - 1)² - 2(x - 1)(x + 2) + (x + 2)² + 5(2x - 3)

• Expand the squares:
  • (x - 1)² = x² - 2x + 1
  • (x + 2)² = x² + 4x + 4
• Simplify the remaining terms:
  • -2(x - 1)(x + 2) = -2(x² + x - 2) = -2x² - 2x + 4
  • 5(2x - 3) = 10x - 15

Combine all terms:

x² - 2x + 1 - 2x² - 2x + 4 + x² + 4x + 4 + 10x - 15

Combine like terms:

(x² - 2x² + x²) + (-2x - 2x + 4x + 10x) + (1 + 4 + 4 - 15)

Simplified Expression:

10x - 6

Conclusion:

The simplified form of the given expression is 10x - 6. By recognizing the pattern and applying the expansion of a squared binomial, we were able to break down the expression and simplify it effectively.