2021 AMC 10A Problems/Problem 1: Difference between revisions
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Knowing that <math>\sqrt{2} \approx 1.41</math> and <math>\sqrt{3} \approx 1.73,</math> we get <cmath>(2-\sqrt{2})(2+\sqrt{2}) - (3-\sqrt{3})(3+\sqrt{3}) + (4-2)(4+2) \approx 0.59\cdot 3.41 -1.26\cdot 4.73 + 2 \cdot 6 | Knowing that <math>\sqrt{2} \approx 1.41</math> and <math>\sqrt{3} \approx 1.73,</math> we get <cmath>(2-\sqrt{2})(2+\sqrt{2}) - (3-\sqrt{3})(3+\sqrt{3}) + (4-2)(4+2) \approx 0.59\cdot 3.41 -1.26\cdot 4.73 + 2 \cdot 6 | ||
=8.0521,</cmath> which is closest to <math>\boxed{\textbf{(D) } 8}.</math> | =8.0521,</cmath> which is closest to <math>\boxed{\textbf{(D) } 8}.</math> | ||
==Video Solution 1 (Lightning Fast)== | ==Video Solution 1 (Lightning Fast)== | ||
Latest revision as of 12:34, 21 September 2025
Problem
What is the value of ![\[(2^2-2)-(3^2-3)+(4^2-4)\]](http://latex.artofproblemsolving.com/f/b/f/fbf9a3889b146c23eb653b9dc16255c03a229dfe.png)
Solution 1
This corresponds to 
-happykeeper
Solution 2
We have
~MRENTHUSIASM
Solution 3 (Overkill: Just for Fun)
We have
-PureSwag
Solution 4 (When you have too much time)
Using the difference of squares, we have ![\[(2-\sqrt{2})(2+\sqrt{2}) - (3-\sqrt{3})(3+\sqrt{3}) + (4-2)(4+2).\]](http://latex.artofproblemsolving.com/d/c/e/dce654c8a197d06312c02b9cf3b07066856347c3.png)
Knowing that 
and 
we get ![\[(2-\sqrt{2})(2+\sqrt{2}) - (3-\sqrt{3})(3+\sqrt{3}) + (4-2)(4+2) \approx 0.59\cdot 3.41 -1.26\cdot 4.73 + 2 \cdot 6 =8.0521,\]](http://latex.artofproblemsolving.com/9/2/c/92c78ba7e8054466f98fc098f06c66d0c2ab6367.png)
which is closest to
Video Solution 1 (Lightning Fast)
~ Education, the Study of everything
Video Solution 3 (Very Very Quick Computation)
https://www.youtube.com/watch?v=m0_UMI2mnZs&list=PLexHyfQ8DMuKqltG3cHT7Di4jhVl6L4YJ&index=2 ~North America Mathematic Contest Go Go Go
Video Solution 4 (Quick Computation)
https://youtu.be/C3n2hgBhyXc?t=37 ~ThePuzzlr
Video Solution 5 by OmegaLearn (Arithmetic Computation)
~ pi_is_3.14
Video Solution 6
~savannahsolver
Video Solution 7
~IceMatrix
Video Solution 8 (Problems 1-3)
~MathWithPi
Video Solution 9
~TheLearningRoyal
See Also
| 2021 AMC 10A (Problems • Answer Key • Resources) | ||
| Preceded by First Problem |
Followed by Problem 2 | |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | ||
| All AMC 10 Problems and Solutions | ||
These problems are copyrighted © by the Mathematical Association of America. 
