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5\left(n^{2}+2n+4\right)
Factor out 5. Polynomial n^{2}+2n+4 is not factored since it does not have any rational roots.
5n^{2}+10n+20=0
Quadratic polynomial can be factored using the transformation ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right), where x_{1} and x_{2} are the solutions of the quadratic equation ax^{2}+bx+c=0.
n=\frac{-10±\sqrt{10^{2}-4\times 5\times 20}}{2\times 5}
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{-10±\sqrt{100-4\times 5\times 20}}{2\times 5}
Square 10.
n=\frac{-10±\sqrt{100-20\times 20}}{2\times 5}
Multiply -4 times 5.
n=\frac{-10±\sqrt{100-400}}{2\times 5}
Multiply -20 times 20.
n=\frac{-10±\sqrt{-300}}{2\times 5}
Add 100 to -400.
5n^{2}+10n+20
Since the square root of a negative number is not defined in the real field, there are no solutions. Quadratic polynomial cannot be factored.
x ^ 2 +2x +4 = 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.This is achieved by dividing both sides of the equation by 5
r + s = -2 rs = 4
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 = -1 - u s = -1 + u
Two numbers r and s sum up to -2 exactly when the average of the two numbers is \frac{1}{2}*-2 = -1. 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>
(-1 - u) (-1 + u) = 4
To solve for unknown quantity u, substitute these in the product equation rs = 4
1 - u^2 = 4
Simplify by expanding (a -b) (a + b) = a^2 – b^2
-u^2 = 4-1 = 3
Simplify the expression by subtracting 1 on both sides
u^2 = -3 u = \pm\sqrt{-3} = \pm \sqrt{3}i
Simplify the expression by multiplying -1 on both sides and take the square root to obtain the value of unknown variable u
r =-1 - \sqrt{3}i s = -1 + \sqrt{3}i
The factors r and s are the solutions to the quadratic equation. Substitute the value of u to compute the r and s.