Track / Stage 4 / 230 of 340 #490 of 1964

Problem 490

AMC 12 late, AIME early

Algebra Difficulty 4.9 Find the answer

1. Find all solutions to the equation $(x-|x|)^{2}+2020(x+|x|)=2020$ .

Your answer

A number or a short expression. Spacing, $ signs and \frac vs / are all fine.

Official solution

Answer: $\{0.5 ;-\sqrt{505}\}$ .

## Solution:

1) Let $x \geq 0$ , then $(x-x)^{2}+2020(x+x)=2020$ , we get $x=0.5$ .
2) Let $x<0, \quad$ then $(x+x)^{2}+2020(x-x)=2020$ , we get $4 x^{2}=2020, \quad x=$ $\pm \sqrt{505}$ , but since $x<0$ , then $x=-\sqrt{505}$ .

Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic, difficulty and ordering added by this site.