Question

How do we solve ${x^2} - 2x + 1 = 0$ by completing the square?

Answer 1

According to the question, first we will arrange the equations and then start removing the constant by adding or subtracting some other constant on both sides.

Complete step by step answer:
The given equation is as:
${x^2} - 2x + 1 = 0$

Now, we will reorder the terms of equation on the basis of their degree,
$1 - 2x + {x^2} = 0$

Begin completing the square, solving for variable $x$ :

Move the constant term to the right side and Add -1 to each side of the equation:
$1 - 2x + ( - 1) + {x^2} = 0 + ( - 1)$

Again, reorder the terms of above equation-
$1 + ( - 1) - 2x + {x^2} = 0 + ( - 1)$

So, 1 and -1 cancel out in the left hand side –
$- 2x + {x^2} = 0 + ( - 1)$
$\because - 2x + {x^2} = - 1$

The $x$ term is 2x. Take half its coefficient (1).
Square it (1) and add it to both sides.
Add ‘1’ to each side of the equation.
$- 2x + 1 + {x^2} = - 1 + 1$

Again reorder the terms-
$1 - 2x + {x^2} = - 1 + 1$

Combine the like terms: $- 1 + 1 = 0$
$1 - 2x + {x^2} = 0$

Factor a perfect square on the left side:
$(x - 1)(x - 1) = 0$

Now, calculate the square root by each of the factors separately.
Break this problem into two cases by setting $(x - 1)$ equal to 0 and 0.

In case-1: $(x - 1) = 0$
$\Rightarrow x = 1$

In case-2: $(x - 1) = 0$
$\Rightarrow x = 1$

The solution to the question is based on the solutions from the cases.
$x = \\{ 1,1\\}$

Note:- There are other methods to compute square root by Long Division Method. Any number can be communicated as a result of prime numbers. This strategy for portrayal of a number as far as the result of prime numbers is named as prime factorization method. It is the most straightforward technique known for the manual computation of the square base of a number.