Difference between revisions of "2020 AMC 10A Problems/Problem 5"

Revision as of 16:07, 31 March 2021

Problem

What is the sum of all real numbers $x$ for which $|x^2-12x+34|=2?$

$\textbf{(A) } 12 \qquad \textbf{(B) } 15 \qquad \textbf{(C) } 18 \qquad \textbf{(D) } 21 \qquad \textbf{(E) } 25$

Solution 1

Split the equation into two cases, where the value inside the absolute value is positive and nonpositive.

Case 1:

The equation yields $x^2-12x+34=2$ , which is equal to $(x-4)(x-8)=0$ . Therefore, the two values for the positive case is $4$ and $8$ .

Case 2:

Similarly, taking the nonpositive case for the value inside the absolute value notation yields $-x^2+12x-34=2$ . Factoring and simplifying gives $(x-6)^2=0$ , so the only value for this case is $6$ .

Summing all the values results in $4+8+6=\boxed{\textbf{(C) }18}$ .

Solution 2

We have the equations $x^2-12x+32=0$ and $x^2-12x+36=0$ .

Notice that the second is a perfect square with a double root at $x=6$ , and the first has two distinct real roots. By Vieta's, the sum of the roots of the first equation is $-(-12)$ or $12$ . $12+6=\boxed{\textbf{(C) }18}$ .

Video Solution 1

https://youtu.be/E7zjQkZl59E

Video Solution 2

Education, The Study Of Everything

https://youtu.be/WUcbVNy2uv0

~IceMatrix

Video Solution 3

https://www.youtube.com/watch?v=7-3sl1pSojc

~bobthefam

Video Solution 4

https://youtu.be/TlIrYXcEuws

~savannahsolver

Video Solution 5

https://youtu.be/3dfbWzOfJAI?t=1544

~ pi_is_3.14

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.

@@ Line 24: / Line 24: @@
 We have the equations <math>x^2-12x+32=0</math> and <math>x^2-12x+36=0</math>.
-Notice that the second is a perfect square with a double root at <math>x=6</math>, and the first has real roots. By Vieta's, the sum of the roots of the first equation is <math>12</math>. <math>12+6=\boxed{\textbf{(C) }18}</math>.
+Notice that the second is a perfect square with a double root at <math>x=6</math>, and the first has two distinct real roots. By Vieta's, the sum of the roots of the first equation is <math>-(-12)</math> or <math>12</math>. <math>12+6=\boxed{\textbf{(C) }18}</math>.
 ==Video Solution 1==

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