Difference between revisions of "2020 AMC 10A Problems/Problem 5"
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== Solution 1== | == Solution 1== | ||
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Split the equation into two cases, where the value inside the absolute value is positive and nonpositive. | Split the equation into two cases, where the value inside the absolute value is positive and nonpositive. | ||
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Case 1: | Case 1: | ||
The equation yields <math>x^2-12x+34=2</math>, which is equal to <math>(x-4)(x-8)=0</math>. Therefore, the two values for the positive case is <math>4</math> and <math>8</math>. | The equation yields <math>x^2-12x+34=2</math>, which is equal to <math>(x-4)(x-8)=0</math>. Therefore, the two values for the positive case is <math>4</math> and <math>8</math>. | ||
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Case 2: | Case 2: | ||
Similarly, taking the nonpositive case for the value inside the absolute value notation yields <math>-x^2+12x-34=2</math>. Factoring and simplifying gives <math>(x-6)^2=0</math>, so the only value for this case is <math>6</math>. | Similarly, taking the nonpositive case for the value inside the absolute value notation yields <math>-x^2+12x-34=2</math>. Factoring and simplifying gives <math>(x-6)^2=0</math>, so the only value for this case is <math>6</math>. | ||
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Summing all the values results in <math>4+8+6=\boxed{\textbf{(C) }18}</math>. | Summing all the values results in <math>4+8+6=\boxed{\textbf{(C) }18}</math>. | ||
Revision as of 23:03, 1 May 2021
Contents
Problem
What is the sum of all real numbers 
for which 
Solution 1
Split the equation into two cases, where the value inside the absolute value is positive and nonpositive.
Case 1:
The equation yields 
, which is equal to 
. Therefore, the two values for the positive case is 
and 
.
Case 2:
Similarly, taking the nonpositive case for the value inside the absolute value notation yields 
. Factoring and simplifying gives 
, so the only value for this case is 
.
Summing all the values results in 
.
Solution 2
We have the equations 
and 
.
Notice that the second is a perfect square with a double root at 
, and the first has two distinct real roots. By Vieta's, the sum of the roots of the first equation is 
or 
. 
.
Video Solution 1
Video Solution 2
Education, The Study Of Everything
~IceMatrix
Video Solution 3
https://www.youtube.com/watch?v=7-3sl1pSojc
~bobthefam
Video Solution 4
~savannahsolver
Video Solution 5
https://youtu.be/3dfbWzOfJAI?t=1544
~ pi_is_3.14
See Also
| 2020 AMC 10A (Problems • Answer Key • Resources) | ||
| Preceded by Problem 4 |
Followed by Problem 6 | |
| 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 | ||
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
