2021 AMC 10A Problems/Problem 1: Difference between revisions

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Problem

What is the value of $(2^2-2)-(3^2-3)+(4^2-4)$

$\textbf{(A) } 1 \qquad\textbf{(B) } 2 \qquad\textbf{(C) } 5 \qquad\textbf{(D) } 8 \qquad\textbf{(E) } 12$

Solution 1

$(4-2)-(9-3)+(16-4)=2-6+12=8.$ This corresponds to $\boxed{\textbf{(D) } 8}.$

-happykeeper

Solution 2

We have $\begin{align*} \left(2^2-2\right)-\left(3^2-3\right)+\left(4^2-4\right) &= 2(2-1)-3(3-1)+4(4-1) \\ &= 2(1)-3(2)+4(3) \\ &= 2-6+12 \\ &= \boxed{\textbf{(D) } 8}. \end{align*}$ ~MRENTHUSIASM

Solution 3 (Overkill: Just for Fun)

We have $\begin{align*} (2^2-2)-(3^2-3)+(4^2-4) &= 2^2+4^2-3^2-2+3-4 \\ &=2^2+(4-3)(4+3)-3 \\ &=2^2+7-3=2^2+4 \\ &=4\cdot 2 \\ &=\boxed{\textbf{(D) } 8}. \end{align*}$ -PureSwag

Solution 4 (When you have too much time)

Using the difference of squares, we have $(2-\sqrt{2})(2+\sqrt{2}) - (3-\sqrt{3})(3+\sqrt{3}) + (4-2)(4+2).$ Knowing that $\sqrt{2} \approx 1.41$ and $\sqrt{3} \approx 1.73,$ we get $(2-\sqrt{2})(2+\sqrt{2}) - (3-\sqrt{3})(3+\sqrt{3}) + (4-2)(4+2) \approx 0.59\cdot 3.41 -1.26\cdot 4.73 + 2 \cdot 6 =8.0521,$ which is closest to $\boxed{\textbf{(D) } 8}.$

Video Solution 1 (Lightning Fast)

https://youtu.be/cAMg15KhK6E

~ 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)

https://youtu.be/0VvM_f-IDvE

~ pi_is_3.14

Video Solution 6

https://youtu.be/4dkzuRHJieQ

~savannahsolver

Video Solution 7

https://youtu.be/50CThrk3RcM

~IceMatrix

Video Solution 8 (Problems 1-3)

https://youtu.be/CupJpUzKPB0

~MathWithPi

Video Solution 9

https://youtu.be/slVBYmcDMOI

~TheLearningRoyal

@@ Line 5: / Line 5: @@
 == Solution 1==
-<math>(4-2)-(9-3)+(16-4)=2-6+12=8.</math> This corresponds to <math>\boxed{\textbf{(D) } 8}</math>
+<math>(4-2)-(9-3)+(16-4)=2-6+12=8.</math> This corresponds to <math>\boxed{\textbf{(D) } 8}.</math>
 -happykeeper
@@ Line 19: / Line 19: @@
 ~MRENTHUSIASM
-==Video Solution 1 (Lightening Fast)==
+==Solution 3 (Overkill: Just for Fun)==
+We have
+<cmath>\begin{align*}
+(2^2-2)-(3^2-3)+(4^2-4) &= 2^2+4^2-3^2-2+3-4 \\
+&=2^2+(4-3)(4+3)-3 \\
+&=2^2+7-3=2^2+4 \\
+&=4\cdot 2 \\
+&=\boxed{\textbf{(D) } 8}.
+\end{align*}</cmath>
+-[[User:PureSwag|PureSwag]]
+==Solution 4 (When you have too much time) ==
+Using the difference of squares, we have <cmath>(2-\sqrt{2})(2+\sqrt{2}) - (3-\sqrt{3})(3+\sqrt{3}) + (4-2)(4+2).</cmath>
+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>
+==Video Solution 1 (Lightning Fast)==
 https://youtu.be/cAMg15KhK6E
 ~ Education, the Study of everything
-==Video Solution 2==
-https://youtu.be/cAMg15KhK6E
 ==Video Solution 3 (Very Very Quick Computation) ==
@@ Line 35: / Line 49: @@
 ~ThePuzzlr
-== Video Solution 5 (Arithmetic Computation) ==
+== Video Solution 5 by OmegaLearn (Arithmetic Computation) ==
 https://youtu.be/0VvM_f-IDvE

2021 AMC 10A (Problems • Answer Key • Resources)
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