Solve for n
n=2
n=6
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n^{2}-8n+12=0
Rearrange the polynomial to put it in standard form. Place the terms in order from highest to lowest power.
a+b=-8 ab=12
To solve the equation, factor n^{2}-8n+12 using formula n^{2}+\left(a+b\right)n+ab=\left(n+a\right)\left(n+b\right). To find a and b, set up a system to be solved.
-1,-12 -2,-6 -3,-4
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 12.
-1-12=-13 -2-6=-8 -3-4=-7
Calculate the sum for each pair.
a=-6 b=-2
The solution is the pair that gives sum -8.
\left(n-6\right)\left(n-2\right)
Rewrite factored expression \left(n+a\right)\left(n+b\right) using the obtained values.
n=6 n=2
To find equation solutions, solve n-6=0 and n-2=0.
n^{2}-8n+12=0
Rearrange the polynomial to put it in standard form. Place the terms in order from highest to lowest power.
a+b=-8 ab=1\times 12=12
To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as n^{2}+an+bn+12. To find a and b, set up a system to be solved.
-1,-12 -2,-6 -3,-4
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 12.
-1-12=-13 -2-6=-8 -3-4=-7
Calculate the sum for each pair.
a=-6 b=-2
The solution is the pair that gives sum -8.
\left(n^{2}-6n\right)+\left(-2n+12\right)
Rewrite n^{2}-8n+12 as \left(n^{2}-6n\right)+\left(-2n+12\right).
n\left(n-6\right)-2\left(n-6\right)
Factor out n in the first and -2 in the second group.
\left(n-6\right)\left(n-2\right)
Factor out common term n-6 by using distributive property.
n=6 n=2
To find equation solutions, solve n-6=0 and n-2=0.
n^{2}-8n+12=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.
n=\frac{-\left(-8\right)±\sqrt{\left(-8\right)^{2}-4\times 12}}{2}
This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -8 for b, and 12 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
n=\frac{-\left(-8\right)±\sqrt{64-4\times 12}}{2}
Square -8.
n=\frac{-\left(-8\right)±\sqrt{64-48}}{2}
Multiply -4 times 12.
n=\frac{-\left(-8\right)±\sqrt{16}}{2}
Add 64 to -48.
n=\frac{-\left(-8\right)±4}{2}
Take the square root of 16.
n=\frac{8±4}{2}
The opposite of -8 is 8.
n=\frac{12}{2}
Now solve the equation n=\frac{8±4}{2} when ± is plus. Add 8 to 4.
n=6
Divide 12 by 2.
n=\frac{4}{2}
Now solve the equation n=\frac{8±4}{2} when ± is minus. Subtract 4 from 8.
n=2
Divide 4 by 2.
n=6 n=2
The equation is now solved.
n^{2}-8n+12=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.
n^{2}-8n+12-12=-12
Subtract 12 from both sides of the equation.
n^{2}-8n=-12
Subtracting 12 from itself leaves 0.
n^{2}-8n+\left(-4\right)^{2}=-12+\left(-4\right)^{2}
Divide -8, the coefficient of the x term, by 2 to get -4. Then add the square of -4 to both sides of the equation. This step makes the left hand side of the equation a perfect square.
n^{2}-8n+16=-12+16
Square -4.
n^{2}-8n+16=4
Add -12 to 16.
\left(n-4\right)^{2}=4
Factor n^{2}-8n+16. 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(n-4\right)^{2}}=\sqrt{4}
Take the square root of both sides of the equation.
n-4=2 n-4=-2
Simplify.
n=6 n=2
Add 4 to both sides of the equation.
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