Question

He sum of the real roots of the equation x2−22013x+∣∣x−22012∣∣+2(24023−1)=0x2−22013x+|x−22012|+2(24023−1)=0 is 2k then k is ____

Answer 1

Answer:

Let f(x) = |x-2|f(x)=∣x−2∣

For x > 2 , f(x) = x-2x>2,f(x)=x−2

for x< 2 , f(x) = 2-xx<2,f(x)=2−x

Thus, for x > 2x>2 equation becomes (x-2)^2 +(x-2) -2 =0(x−2)

2

+(x−2)−2=0

x^2 - 3x = 0x

2

−3x=0

Thus, the root of equation x > 2x>2 is 33

For x < 2x<2 the equation becomes (x-2)^2 + (2-x) -2 =0(x−2)

2

+(2−x)−2=0

x^2 - 5x + 4 =0x

2

−5x +4=0

(x-4)(x-1) = 0(x−4)(x−1)=0

Root which is less than 22 is 11

Thus, roots of given equations are 33, 11

Sum will be 3+1 =43+1=4