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 $ax^{2}+bx+c=0$.
$$4n^{2}-78n-360=0$$
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.
Now solve the equation $n=\frac{78±6\sqrt{329}}{8}$ when $±$ is plus. Add $78$ to $6\sqrt{329}$.
$$n=\frac{6\sqrt{329}+78}{8}$$
Divide $78+6\sqrt{329}$ by $8$.
$$n=\frac{3\sqrt{329}+39}{4}$$
Now solve the equation $n=\frac{78±6\sqrt{329}}{8}$ when $±$ is minus. Subtract $6\sqrt{329}$ from $78$.
$$n=\frac{78-6\sqrt{329}}{8}$$
Divide $78-6\sqrt{329}$ by $8$.
$$n=\frac{39-3\sqrt{329}}{4}$$
Factor the original expression using $ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right)$. Substitute $\frac{39+3\sqrt{329}}{4}$ for $x_{1}$ and $\frac{39-3\sqrt{329}}{4}$ for $x_{2}$.
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 $4$
$$x ^ 2 -\frac{39}{2}x -90 = 0$$
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 $rs = C$
$$r + s = \frac{39}{2} $$ $$ rs = -90$$
Two numbers $r$ and $s$ sum up to $\frac{39}{2}$ exactly when the average of the two numbers is $\frac{1}{2}*\frac{39}{2} = \frac{39}{4}$. 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 $u$.
$$r = \frac{39}{4} - u$$ $$s = \frac{39}{4} + u$$
To solve for unknown quantity $u$, substitute these in the product equation $rs = -90$
$$(\frac{39}{4} - u) (\frac{39}{4} + u) = -90$$
Simplify by expanding $(a -b) (a + b) = a^2 – b^2$
$$\frac{1521}{16} - u^2 = -90$$
Simplify the expression by subtracting $\frac{1521}{16}$ on both sides
$$-u^2 = -90-\frac{1521}{16} = -\frac{2961}{16}$$
Simplify the expression by multiplying $-1$ on both sides and take the square root to obtain the value of unknown variable $u$
