2022 AMC 8 Problems/Problem 2: Difference between revisions
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==Solution 2(Easier)== | |||
First, let's tackle the <math>\blacklozenge</math> part since it is in brackets. For the <math>\blacklozenge</math> part, the formula is - | |||
<cmath> (a \, \blacklozenge \, b) = a^2 - b^2</cmath> | |||
Substituting <math>5</math> and <math>3</math> gives us - | |||
<cmath> 5^2 - 3^2 = 16</cmath> | |||
Now onto the <math>\bigstar</math> part. For the <math>\bigstar part, the formula is - | |||
<cmath> (a \, \bigstar \, b) = (a - b)^2</cmath> | |||
Substituting the values </math>16<math>, which we just got, and </math>6$, gives us - | |||
<cmath> (16 - 6)^2 = </cmath> | |||
<cmath> (10)^2 =</cmath> | |||
<cmath> =\boxed{\textbf{(D) } 100} </cmath> | |||
~sdmd | |||
==Video Solution (A Clever Explanation You’ll Get Instantly)== | ==Video Solution (A Clever Explanation You’ll Get Instantly)== | ||
Revision as of 11:37, 20 January 2025
Problem
Consider these two operations:
What is the output of 
Solution 1
We can find a general solution to any 
.
![\[((a \, \blacklozenge \, b) \, \bigstar \, c)\]](http://latex.artofproblemsolving.com/9/b/1/9b1ead4edbaae012d71fc60a0fb5f97d60ea4a95.png)
![\[=((a^2-b^2) \, \bigstar \, c)\]](http://latex.artofproblemsolving.com/1/3/0/130283c842ecd4a186bfc2a3a91da5d10b49d92e.png)
![\[=(a^2-b^2-c)^2\]](http://latex.artofproblemsolving.com/0/c/a/0ca0fb3ae428092c4149b4c4afdd2bb3bb2306e0.png)
![\[=a^4+b^4-(a^2)(b^2)-2(a^2)(c)-(b^2)(a^2)+2(b^2)(c)+c^2\]](http://latex.artofproblemsolving.com/0/0/d/00dc98219662e298a83f59b9e87e169d34ef4dc5.png)
![\[=5^4+3^4-(5^2)(3^2)-2(5^2)(6)-(3^2)(5^2)+2(3^2)(6)+6^2\]](http://latex.artofproblemsolving.com/4/e/d/4edeffd78b5b1feb3e9fdf7af355cff8ed2829a1.png)
![\[=625+81-225-300-225+108+36\]](http://latex.artofproblemsolving.com/2/3/5/2358711b9bc73279606f6b4824474e91e457133a.png)
![\[=\boxed{\textbf{(D) } 100}\]](http://latex.artofproblemsolving.com/c/a/d/cad27560c19ca35edc6eeac0171d9866b590ad60.png)
~megaboy6679
Solution 2(Easier)
First, let's tackle the 
part since it is in brackets. For the part, the formula is -
![\[(a \, \blacklozenge \, b) = a^2 - b^2\]](http://latex.artofproblemsolving.com/d/7/e/d7e8d5fcda200566180ecb08a90a55a760425c67.png)
Substituting 
and 
gives us -
![\[5^2 - 3^2 = 16\]](http://latex.artofproblemsolving.com/6/9/2/692f182bbd0270b311899e92d87d8d9d3c5283e0.png)
Now onto the 
part. For the 
16
6$, gives us -
![\[(16 - 6)^2 =\]](http://latex.artofproblemsolving.com/3/0/e/30ec6dee4216a2d12cb1f908f8e503f4c85b84ae.png)
![\[(10)^2 =\]](http://latex.artofproblemsolving.com/d/1/c/d1c907c1c2c5f16917f830857a527006982261cb.png)
~sdmd
Video Solution (A Clever Explanation You’ll Get Instantly)
https://youtu.be/tYWp6fcUAik?si=V8hv_zOn_zYOi9E5&t=74 ~hsnacademy
Video Solution 1 by Math-X (First understand the problem!!!)
https://youtu.be/oUEa7AjMF2A?si=JlQdlwkdbIYFNJXj&t=123
~Math-X
Video Solution 2 (HOW TO THINK CREATIVELY!!!)
~Education, the Study of Everything
Video Solution 3
https://www.youtube.com/watch?v=Ij9pAy6tQSg&t=91 ~Interstigation
Video Solution 4
~savannahsolver
Video Solution 5
https://youtu.be/Q0R6dnIO95Y?t=53
~STEMbreezy
Video Solution 6
https://www.youtube.com/watch?v=c123kPqd11I
~harungurcan
Video Solution 7 by Dr. David
See Also
| 2022 AMC 8 (Problems • Answer Key • Resources) | ||
| Preceded by Problem 1 |
Followed by Problem 3 | |
| 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 AJHSME/AMC 8 Problems and Solutions | ||
These problems are copyrighted © by the Mathematical Association of America, as part of the American Mathematics Competitions. Error creating thumbnail: File missing
