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x^{2}-6x+9-8\left(-x+1\right)=0
Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x-3\right)^{2}.
x^{2}-6x+9-8\left(-x\right)-8=0
Use the distributive property to multiply -8 by -x+1.
x^{2}-6x+9+8x-8=0
Multiply -8 and -1 to get 8.
x^{2}+2x+9-8=0
Combine -6x and 8x to get 2x.
x^{2}+2x+1=0
Subtract 8 from 9 to get 1.
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 positive, a and b are both positive. 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}-6x+9-8\left(-x+1\right)=0
Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x-3\right)^{2}.
x^{2}-6x+9-8\left(-x\right)-8=0
Use the distributive property to multiply -8 by -x+1.
x^{2}-6x+9+8x-8=0
Multiply -8 and -1 to get 8.
x^{2}+2x+9-8=0
Combine -6x and 8x to get 2x.
x^{2}+2x+1=0
Subtract 8 from 9 to get 1.
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 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.
\left(x+1\right)^{2}
Rewrite as a binomial square.
x=-1
To find equation solution, solve x+1=0.
x^{2}-6x+9-8\left(-x+1\right)=0
Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x-3\right)^{2}.
x^{2}-6x+9-8\left(-x\right)-8=0
Use the distributive property to multiply -8 by -x+1.
x^{2}-6x+9+8x-8=0
Multiply -8 and -1 to get 8.
x^{2}+2x+9-8=0
Combine -6x and 8x to get 2x.
x^{2}+2x+1=0
Subtract 8 from 9 to get 1.
x=\frac{-2±\sqrt{2^{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{-2±\sqrt{4-4}}{2}
Square 2.
x=\frac{-2±\sqrt{0}}{2}
Add 4 to -4.
x=-\frac{2}{2}
Take the square root of 0.
x=-1
Divide -2 by 2.
x^{2}-6x+9-8\left(-x+1\right)=0
Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x-3\right)^{2}.
x^{2}-6x+9-8\left(-x\right)-8=0
Use the distributive property to multiply -8 by -x+1.
x^{2}-6x+9+8x-8=0
Multiply -8 and -1 to get 8.
x^{2}+2x+9-8=0
Combine -6x and 8x to get 2x.
x^{2}+2x+1=0
Subtract 8 from 9 to get 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
Subtract 1 from both sides of the equation.
x=-1
The equation is now solved. Solutions are the same.