Question

Find the sum of all real roots of the equation (x - 2) ^ 2 + |x - 2| - 2 = 0​

Answer 1

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Let f(x)=∣x−2∣

For x>2,f(x)=x−2

for x<2,f(x)=2−x

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

x^2−3x=0

Thus, the root of equation x>2 is 3

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

x^2−5x+4=0

(x−4)(x−1)=0

Root which is less than 2 is 1

Thus, roots of given equations are 3, 1

Sum will be 3+1=4

Answer 2

Answer:

Let f(x)=∣x−2∣

For x>2,f(x)=x−2

for x<2,f(x)=2−x

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

x^2−3x=0

Thus, the root of equation x>2 is 3

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

x^2−5x+4=0

(x−4)(x−1)=0

Root which is less than 2 is 1

Thus, roots of given equations are 3, 1

Sum will be 3+1=4