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x^{2}-16x+64=0
Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x-8\right)^{2}.
a+b=-16 ab=64
To solve the equation, factor x^{2}-16x+64 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.
-1,-64 -2,-32 -4,-16 -8,-8
Since ab is positive, a and b have the same sign. Since a+b is negative, a and b are both negative. List all such integer pairs that give product 64.
-1-64=-65 -2-32=-34 -4-16=-20 -8-8=-16
Calculate the sum for each pair.
a=-8 b=-8
The solution is the pair that gives sum -16.
\left(x-8\right)\left(x-8\right)
Rewrite factored expression \left(x+a\right)\left(x+b\right) using the obtained values.
\left(x-8\right)^{2}
Rewrite as a binomial square.
x=8
To find equation solution, solve x-8=0.
x^{2}-16x+64=0
Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x-8\right)^{2}.
a+b=-16 ab=1\times 64=64
To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as x^{2}+ax+bx+64. To find a and b, set up a system to be solved.
-1,-64 -2,-32 -4,-16 -8,-8
Since ab is positive, a and b have the same sign. Since a+b is negative, a and b are both negative. List all such integer pairs that give product 64.
-1-64=-65 -2-32=-34 -4-16=-20 -8-8=-16
Calculate the sum for each pair.
a=-8 b=-8
The solution is the pair that gives sum -16.
\left(x^{2}-8x\right)+\left(-8x+64\right)
Rewrite x^{2}-16x+64 as \left(x^{2}-8x\right)+\left(-8x+64\right).
x\left(x-8\right)-8\left(x-8\right)
Factor out x in the first and -8 in the second group.
\left(x-8\right)\left(x-8\right)
Factor out common term x-8 by using distributive property.
\left(x-8\right)^{2}
Rewrite as a binomial square.
x=8
To find equation solution, solve x-8=0.
x^{2}-16x+64=0
Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x-8\right)^{2}.
x=\frac{-\left(-16\right)±\sqrt{\left(-16\right)^{2}-4\times 64}}{2}
This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -16 for b, and 64 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
x=\frac{-\left(-16\right)±\sqrt{256-4\times 64}}{2}
Square -16.
x=\frac{-\left(-16\right)±\sqrt{256-256}}{2}
Multiply -4 times 64.
x=\frac{-\left(-16\right)±\sqrt{0}}{2}
Add 256 to -256.
x=-\frac{-16}{2}
Take the square root of 0.
x=\frac{16}{2}
The opposite of -16 is 16.
x=8
Divide 16 by 2.
\sqrt{\left(x-8\right)^{2}}=\sqrt{0}
Take the square root of both sides of the equation.
x-8=0 x-8=0
Simplify.
x=8 x=8
Add 8 to both sides of the equation.
x=8
The equation is now solved. Solutions are the same.