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

(Created page with "==See Also== {{AMC10 box|year=2020|ab=A|num-b=4|num-a=6}} {{MAA Notice}}")
 
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==Problem 5==
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What is the sum of all real numbers <math>x</math> for which <math>|x^2-12x+34|=2?</math>
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<math>\textbf{(A) } 12 \qquad \textbf{(B) } 15 \qquad \textbf{(C) } 18 \qquad \textbf{(D) } 21 \qquad \textbf{(E) } 25</math>
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== Solution ==
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Split the equation into two cases, where the value inside the absolute value is positive and nonpositive.
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The first case 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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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{\text{(C) }18}</math>.
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==See Also==
 
==See Also==
  
 
{{AMC10 box|year=2020|ab=A|num-b=4|num-a=6}}
 
{{AMC10 box|year=2020|ab=A|num-b=4|num-a=6}}
 
{{MAA Notice}}
 
{{MAA Notice}}

Revision as of 21:13, 31 January 2020

Problem 5

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

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

The first case 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 $8$.

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 $4+8+6=\boxed{\text{(C) }18}$.

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

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