SATHEE: Theory of Equations 1 Question 38

Theory of Equations 1 Question 38

39. The sum of all the real roots of the equation $| x - 2 |^{2} + | x - 2 | - 2 = 0$ is…… .

(1997, 2M)

Show Answer

Answer:

Correct Answer: 39. (4)

Solution:

Given, $| x - 2 |^{2} + | x - 2 | - 2 = 0$

Case I When $x \geq 2$

$\begin{array}{cc} \Rightarrow & (x - 2)^{2} + (x - 2) - 2 = 0 \\ \Rightarrow & x^{2} + 4 - 4 x + x - 2 - 2 = 0 \\ \Rightarrow & x^{2} - 3 x = 0 \\ \Rightarrow & x (x - 3) = 0 \\ \Rightarrow & x = 0, 3 \\ \Rightarrow & x = 3 \end{array}$

Case II When $x < 2$

$\Rightarrow {- (x - 2)}^{2} - (x - 2) - 2 = 0$

$\Rightarrow (x - 2)^{2} - x + 2 - 2 = 0$

$\Rightarrow x^{2} + 4 - 4 x - x = 0$

$\Rightarrow x^{2} - 4 x - (x - 4) = 0$

$\Rightarrow x (x - 4) - 1 (x - 4) = 0$

$\Rightarrow (x - 1) (x - 4) = 0$

$\begin{array}{ll} \Rightarrow & x = 1, 4 \\ \Rightarrow & x = 1 \end{array}$

[4 is rejected]

Hence, the sum of the roots is $3 + 1 = 4$ .

Alternate Solution

Given, $| x - 2 |^{2} + | x - 2 | - 2 = 0$

$\Rightarrow (| x - 2 | + 2) (| x - 2 | - 1) = 0$

$∴ | x - 2 | = - 2, 1$ [neglecting -2 ]

$\Rightarrow | x - 2 | = 1 \Rightarrow x = 3, 1$

$\Rightarrow$ Sum of the roots $= 4$

Theory of Equations 1 Question 38

39. The sum of all the real roots of the equation $| x - 2 |^{2} + | x - 2 | - 2 = 0$ is…… .

Answer:

Alternate Solution

Engineering Preparation

Medical Preparation

NCERT Books & Solutions

Important Resources

Other Resources

Syllabus

Test Platform

Sample Mock Test

Mentorship

NCERT Exercise Video Solution

Theory of Equations 1 Question 38

39. The sum of all the real roots of the equation |x−2|2+|x−2|−2=0 is…… .

Answer:

Alternate Solution

Engineering Preparation

Medical Preparation

NCERT Books & Solutions

Important Resources

Other Resources

Syllabus

Test Platform

Sample Mock Test

Mentorship

NCERT Exercise Video Solution

Menu

39. The sum of all the real roots of the equation $| x - 2 |^{2} + | x - 2 | - 2 = 0$ is…… .