Question

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

Answer 1

Answer:

Sum of roots = 4

Step-by-step explanation:

Given:

(|x-2|)²+ |x-2| -2=0

Here arises two cases for the variable inside modulus

$\boxed{\bold{\mathsf{Case\:\: 1}}}$

When x ≥ 2

(x-2)² + x- 2 -2=0

x² + 4 -4x x - 2 - 2 = 0

x² - 3x = 0

x(x - 3) = 0

Equate the factors to zero

x = 0

$\rule{100}{1}$

x - 3 = 0

x = 3

•°• we get two roots as 0 , 3 but 0 will be rejected as its not satisfies the condition x ≥ 2 .

$\rule{200}{1}$

★ $\boxed{\bold{\mathsf{Case \:\: 2}}}$

When x < 2

[- (x -2)]² - (x-2) -2 = 0

(x-2)² - x +2 -2 = 0

x² + 4 - 4x - x = 0

x² - 4x - (x -4) = 0

x(x-1) -1 (x-4) = 0

(x -1) (x-4) = 0

Equate the factors to zero

(x -1) = 0

x = 1

$\rule{100}{1}$

(x-4) = 0

x = 4

•°• we get two roots as 1 , 4 but 4 will be rejected as its not satisfies the condition x < 2 .

$\rule{200}{1}$

Thus, Sum of roots is 3 + 1 = 4

$\boxed{\bold{\mathsf{Sum\: of\ roots\ is \: 4 }}}$