Find all real solutions to the equation with absolute value of |x−1| = ∣x2−2x+1∣.

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No Solutions

0, 1, 2

answer is D.

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Detailed Solution

Take the given expression –
Find all real solutions to the equation with absolute value
|x−1| = ∣x2−2x+1∣
It can be re-written as – Moving the left hand side of the equation to the right hand side of the equation and vice-versa.
∣x2−2x+1∣ = |x−1|
To find the real solutions remove mode and take the plus or minus sign on the right hand side of the equation.
(x2−2x+1) = ±(x−1)
Take one the left hand side of the equation to the negative of the right hand side of the equation and the second time positive to the right hand side of the equation.
(x2−2x+1) = +(x−1)
Open the brackets; remember when there is a positive sign outside the bracket then values inside the bracket do not change.
⇒x2−2x+1= x−1
Take all the terms on one side of the equation. Also, when any term is moved from one side to another then the sign also changes. Positive term changes to negative and vice-versa.
⇒x2−2x+1−x+1 = 0
Make a pair of like terms.
⇒x2−2x−x+1+1 = 0
Simplify the above equation –
⇒x2−3x+2 = 0
By factorization method, the equation can be re-written as –
⇒x2−2x-x+2 = 0
Take out the common multiple in the two pairs-
⇒x(x−2)−1(x−2) = 0
Take common multiple out –
⇒(x−2)(x−1) = 0
We get –
⇒x−1=0 or x - 2 = 0
⇒x = 1 or x = 2 ..... (a)
Similarly for the second case-
(x2−2x+1) = −(x−1)
Open the brackets; remember when there is a negative sign outside the bracket then values inside the bracket do change. Positive term changes to negative and vice-versa.
⇒x2−2x+1 = −x+1
Take all the terms on one side of the equation. Also, when any term is moved from one side to another then the sign also changes. Positive term changes to negative and vice-versa.
⇒x2−2x+1+x−1 = 0
Make a pair of like terms.
⇒x2−2x+x+1−1 = 0
Simplify the above equation –
⇒x2−x = 0
Take out the common multiple in the above equation -
⇒x(x−1) = 0
We get –
⇒x−1= 0 or x = 0
⇒x = 1 or x = 0 ..... (b)
Therefore, the real solutions of the given equation are 0, 1 and 2.
So, the correct answer is “Option 4”.

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