2020 AMC 10A Problems/Problem 5

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 1

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

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 real roots. By Vieta's, the sum of the roots of the first equation is $12$ . $12+6=\boxed{\text{(C) }18}$ .

2020 AMC 10A (Problems • Answer Key • Resources)
Preceded by Problem 4	Followed by Problem 6
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2020 AMC 10A Problems/Problem 5

Contents

Problem 5

Solution 1

Solution 2

See Also