AMC 10A 2020 (Problem 5)What is the sum of all real numbers xxx for which ∣x2−12x+34∣=2\left|x^2-12x+34\right|=2x2−12x+34=2?(A) 12\text{(A)}\;12(A)12(B) 15\text{(B)}\;15(B)15(C) 18\text{(C)}\;18(C)18(D) 21\text{(D)}\;21(D)21(E) 25\text{(E)}\;25(E)25Related TopicsCoreToolkit 7 — Square of a differenceToolkit 19 — Factoring quadratic by groupingHints (4)Hint 1x2−12x+34=±2x^2-12x+34=\pm2x2−12x+34=±2Hint 2Case 1:x2−12x+34=2x^2-12x+34=2x2−12x+34=2x2−12x+32=0x^2-12x+32=0x2−12x+32=0(x−4)(x−8)=0(x-4)(x-8)=0(x−4)(x−8)=0⟹x=4,8\Longrightarrow x=4,8⟹x=4,8Hint 3Case 2:x2−12x+34=−2x^2-12x+34=-2x2−12x+34=−2x2−12x+36=0x^2-12x+36=0x2−12x+36=0(x−6)2=0(x-6)^2=0(x−6)2=0⟹x=6\Longrightarrow x=6⟹x=6Hint 4By Hints 2 and 3,Ans=4+8+6=18\text{Ans}=4+8+6=18Ans=4+8+6=18Final Answer(C) 181818Related Problems (3)MATHCOUNTS State Sprint 2025 (Problem 26)BMO1 2016/2017 (Problem 3)AMC 10B 2025 (Problem 15)