Question

The product of all solutions of the equation $(x-2)^2-3|x-2|+2=0$ is
a) 2
b) -4
c) 0
d) None of these

Answer 1

Hint: Consider (x-2) as f(x) and solve the equation. After putting value we will get two quadratic equations. Solve them separately to get the root of the equation . Then find the product of all solutions.

Complete step-by-step answer:
Given , the equation $(x-2)^2-3|x-2|+2=0$ is a quadratic equation
For x>2 , f(x) = x-2
For x<2 , f(x) = 2-x
Thus for x>2 , the equation becomes $(x-2)^2-3(x-2)+2=0$
 Solving the equation we get ,
$(x-2)^2-3(x-2)+2=0$
$\Rightarrow x^2-4x+4-3x+6+2=0$
$\Rightarrow x(x-4)-3(x-4)=0$
$\Rightarrow (x-4)(x-3)=0$
Roots for equation x>2 are 3,4
Similarly , for x<2 , the equation becomes
$(x-2)^2-3(2-x)+2=0$
$\Rightarrow x^2-4x+4+3x-6+2=0$
$\Rightarrow x^2-x=0$
$\Rightarrow x(x-1)=0$
Root for equation x<2 is 0,1
Thus roots of the equation are 3,4 (x>2) and 0,1 (x<2)
Product of the roots are = 0
Hence , the required option is c)0


Note: A linear equation is an equation that may be put in the form where are the variables, and are the coefficients, which are often real numbers. The coefficients may be considered as parameters of the equation. Students often don’t understand the way to solve equations which they can understand better by understanding what linear equations are.