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a+b=-12 ab=27
To solve the equation, factor n^{2}-12n+27 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,-27 -3,-9
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 27.
-1-27=-28 -3-9=-12
Calculate the sum for each pair.
a=-9 b=-3
The solution is the pair that gives sum -12.
\left(n-9\right)\left(n-3\right)
Rewrite factored expression \left(n+a\right)\left(n+b\right) using the obtained values.
n=9 n=3
To find equation solutions, solve n-9=0 and n-3=0.
a+b=-12 ab=1\times 27=27
To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as n^{2}+an+bn+27. To find a and b, set up a system to be solved.
-1,-27 -3,-9
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 27.
-1-27=-28 -3-9=-12
Calculate the sum for each pair.
a=-9 b=-3
The solution is the pair that gives sum -12.
\left(n^{2}-9n\right)+\left(-3n+27\right)
Rewrite n^{2}-12n+27 as \left(n^{2}-9n\right)+\left(-3n+27\right).
n\left(n-9\right)-3\left(n-9\right)
Factor out n in the first and -3 in the second group.
\left(n-9\right)\left(n-3\right)
Factor out common term n-9 by using distributive property.
n=9 n=3
To find equation solutions, solve n-9=0 and n-3=0.
n^{2}-12n+27=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(-12\right)±\sqrt{\left(-12\right)^{2}-4\times 27}}{2}
This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -12 for b, and 27 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
n=\frac{-\left(-12\right)±\sqrt{144-4\times 27}}{2}
Square -12.
n=\frac{-\left(-12\right)±\sqrt{144-108}}{2}
Multiply -4 times 27.
n=\frac{-\left(-12\right)±\sqrt{36}}{2}
Add 144 to -108.
n=\frac{-\left(-12\right)±6}{2}
Take the square root of 36.
n=\frac{12±6}{2}
The opposite of -12 is 12.
n=\frac{18}{2}
Now solve the equation n=\frac{12±6}{2} when ± is plus. Add 12 to 6.
n=9
Divide 18 by 2.
n=\frac{6}{2}
Now solve the equation n=\frac{12±6}{2} when ± is minus. Subtract 6 from 12.
n=3
Divide 6 by 2.
n=9 n=3
The equation is now solved.
n^{2}-12n+27=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}-12n+27-27=-27
Subtract 27 from both sides of the equation.
n^{2}-12n=-27
Subtracting 27 from itself leaves 0.
n^{2}-12n+\left(-6\right)^{2}=-27+\left(-6\right)^{2}
Divide -12, the coefficient of the x term, by 2 to get -6. Then add the square of -6 to both sides of the equation. This step makes the left hand side of the equation a perfect square.
n^{2}-12n+36=-27+36
Square -6.
n^{2}-12n+36=9
Add -27 to 36.
\left(n-6\right)^{2}=9
Factor n^{2}-12n+36. 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-6\right)^{2}}=\sqrt{9}
Take the square root of both sides of the equation.
n-6=3 n-6=-3
Simplify.
n=9 n=3
Add 6 to both sides of the equation.
x ^ 2 -12x +27 = 0
Quadratic equations such as this one can be solved by a new direct factoring method that does not require guess work. To use the direct factoring method, the equation must be in the form x^2+Bx+C=0.
r + s = 12 rs = 27
Let r and s be the factors for the quadratic equation such that x^2+Bx+C=(x−r)(x−s) where sum of factors (r+s)=−B and the product of factors rs = C
r = 6 - u s = 6 + u
Two numbers r and s sum up to 12 exactly when the average of the two numbers is \frac{1}{2}*12 = 6. You can also see that the midpoint of r and s corresponds to the axis of symmetry of the parabola represented by the quadratic equation y=x^2+Bx+C. The values of r and s are equidistant from the center by an unknown quantity u. Express r and s with respect to variable u. <div style='padding: 8px'><img src='https://opalmath.azureedge.net/customsolver/quadraticgraph.png' style='width: 100%;max-width: 700px' /></div>
(6 - u) (6 + u) = 27
To solve for unknown quantity u, substitute these in the product equation rs = 27
36 - u^2 = 27
Simplify by expanding (a -b) (a + b) = a^2 – b^2
-u^2 = 27-36 = -9
Simplify the expression by subtracting 36 on both sides
u^2 = 9 u = \pm\sqrt{9} = \pm 3
Simplify the expression by multiplying -1 on both sides and take the square root to obtain the value of unknown variable u
r =6 - 3 = 3 s = 6 + 3 = 9
The factors r and s are the solutions to the quadratic equation. Substitute the value of u to compute the r and s.