Question

\[ \text{Find } 2011^2 - 2010^2 \text{?} \]

Ask by Bob Harris. in South Africa
Mar 09,2025

Upstudy AI Solution

Tutor-Verified Answer

Answer

\( 2011^2 - 2010^2 = 4021 \)

Solution

1. Recognize the difference of squares formula: \[ a^2 - b^2 = (a-b)(a+b) \] 2. Let \( a = 2011 \) and \( b = 2010 \), then: \[ 2011^2 - 2010^2 = (2011-2010)(2011+2010) \] 3. Calculate the difference: \[ 2011 - 2010 = 1 \] 4. Calculate the sum: \[ 2011 + 2010 = 4021 \] 5. Multiply the results: \[ 1 \times 4021 = 4021 \] Thus, the final answer is \(\boxed{4021}\).

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The Deep Dive

Did you know that we can solve \(2011^2 - 2010^2\) using the difference of squares formula? This nifty formula states that \(a^2 - b^2 = (a-b)(a+b)\). Here, if we let \(a = 2011\) and \(b = 2010\), we get: \[ 2011^2 - 2010^2 = (2011 - 2010)(2011 + 2010) = 1 \times 4021 = 4021. \] It's a great example of how math shortcuts can save time! Now, for a little bit of historical trivia, the difference of squares has been used since ancient times, notably by mathematicians in Babylon. They recognized the pattern and used it for calculations long before the formal "algebra" we know today was developed. Not only does it yield quick results, but it also showcases the timeless nature of mathematical truths!

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