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x^{2}-2x+1=0
Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x-1\right)^{2}.
a+b=-2 ab=1
To solve the equation, factor x^{2}-2x+1 using formula x^{2}+\left(a+b\right)x+ab=\left(x+a\right)\left(x+b\right). To find a and b, set up a system to be solved.
a=-1 b=-1
Since ab is positive, a and b have the same sign. Since a+b is negative, a and b are both negative. The only such pair is the system solution.
\left(x-1\right)\left(x-1\right)
Rewrite factored expression \left(x+a\right)\left(x+b\right) using the obtained values.
\left(x-1\right)^{2}
Rewrite as a binomial square.
x=1
To find equation solution, solve x-1=0.
x^{2}-2x+1=0
Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x-1\right)^{2}.
a+b=-2 ab=1\times 1=1
To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as x^{2}+ax+bx+1. To find a and b, set up a system to be solved.
a=-1 b=-1
Since ab is positive, a and b have the same sign. Since a+b is negative, a and b are both negative. The only such pair is the system solution.
\left(x^{2}-x\right)+\left(-x+1\right)
Rewrite x^{2}-2x+1 as \left(x^{2}-x\right)+\left(-x+1\right).
x\left(x-1\right)-\left(x-1\right)
Factor out x in the first and -1 in the second group.
\left(x-1\right)\left(x-1\right)
Factor out common term x-1 by using distributive property.
\left(x-1\right)^{2}
Rewrite as a binomial square.
x=1
To find equation solution, solve x-1=0.
x^{2}-2x+1=0
Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x-1\right)^{2}.
x=\frac{-\left(-2\right)±\sqrt{\left(-2\right)^{2}-4}}{2}
This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -2 for b, and 1 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
x=\frac{-\left(-2\right)±\sqrt{4-4}}{2}
Square -2.
x=\frac{-\left(-2\right)±\sqrt{0}}{2}
Add 4 to -4.
x=-\frac{-2}{2}
Take the square root of 0.
x=\frac{2}{2}
The opposite of -2 is 2.
x=1
Divide 2 by 2.
\sqrt{\left(x-1\right)^{2}}=\sqrt{0}
Take the square root of both sides of the equation.
x-1=0 x-1=0
Simplify.
x=1 x=1
Add 1 to both sides of the equation.
x=1
The equation is now solved. Solutions are the same.