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-x^{2}+2x-1=0
Divide both sides by 5.
a+b=2 ab=-\left(-1\right)=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 positive, a and b are both positive. 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)+x-1
Factor out -x in -x^{2}+x.
\left(x-1\right)\left(-x+1\right)
Factor out common term x-1 by using distributive property.
x=1 x=1
To find equation solutions, solve x-1=0 and -x+1=0.
-5x^{2}+10x-5=0
All equations of the form ax^{2}+bx+c=0 can be solved using the quadratic formula: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. The quadratic formula gives two solutions, one when ± is addition and one when it is subtraction.
x=\frac{-10±\sqrt{10^{2}-4\left(-5\right)\left(-5\right)}}{2\left(-5\right)}
This equation is in standard form: ax^{2}+bx+c=0. Substitute -5 for a, 10 for b, and -5 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
x=\frac{-10±\sqrt{100-4\left(-5\right)\left(-5\right)}}{2\left(-5\right)}
Square 10.
x=\frac{-10±\sqrt{100+20\left(-5\right)}}{2\left(-5\right)}
Multiply -4 times -5.
x=\frac{-10±\sqrt{100-100}}{2\left(-5\right)}
Multiply 20 times -5.
x=\frac{-10±\sqrt{0}}{2\left(-5\right)}
Add 100 to -100.
x=-\frac{10}{2\left(-5\right)}
Take the square root of 0.
x=-\frac{10}{-10}
Multiply 2 times -5.
x=1
Divide -10 by -10.
-5x^{2}+10x-5=0
Quadratic equations such as this one can be solved by completing the square. In order to complete the square, the equation must first be in the form x^{2}+bx=c.
-5x^{2}+10x-5-\left(-5\right)=-\left(-5\right)
Add 5 to both sides of the equation.
-5x^{2}+10x=-\left(-5\right)
Subtracting -5 from itself leaves 0.
-5x^{2}+10x=5
Subtract -5 from 0.
\frac{-5x^{2}+10x}{-5}=\frac{5}{-5}
Divide both sides by -5.
x^{2}+\frac{10}{-5}x=\frac{5}{-5}
Dividing by -5 undoes the multiplication by -5.
x^{2}-2x=\frac{5}{-5}
Divide 10 by -5.
x^{2}-2x=-1
Divide 5 by -5.
x^{2}-2x+1=-1+1
Divide -2, the coefficient of the x term, by 2 to get -1. Then add the square of -1 to both sides of the equation. This step makes the left hand side of the equation a perfect square.
x^{2}-2x+1=0
Add -1 to 1.
\left(x-1\right)^{2}=0
Factor x^{2}-2x+1. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{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.