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 $ax^{2}+bx+c=0$.
$$x^{2}-200x-24000=0$$
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 $x=\frac{200±40\sqrt{85}}{2}$ when $±$ is plus. Add $200$ to $40\sqrt{85}$.
$$x=\frac{40\sqrt{85}+200}{2}$$
Divide $200+40\sqrt{85}$ by $2$.
$$x=20\sqrt{85}+100$$
Now solve the equation $x=\frac{200±40\sqrt{85}}{2}$ when $±$ is minus. Subtract $40\sqrt{85}$ from $200$.
$$x=\frac{200-40\sqrt{85}}{2}$$
Divide $200-40\sqrt{85}$ by $2$.
$$x=100-20\sqrt{85}$$
Factor the original expression using $ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right)$. Substitute $100+20\sqrt{85}$ for $x_{1}$ and $100-20\sqrt{85}$ 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 $x^2+Bx+C=0$.
$$x ^ 2 -200x -24000 = 0$$
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 $rs = C$
$$r + s = 200 $$ $$ rs = -24000$$
Two numbers $r$ and $s$ sum up to $200$ exactly when the average of the two numbers is $\frac{1}{2}*200 = 100$. 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 $u$.
$$r = 100 - u$$ $$s = 100 + u$$
To solve for unknown quantity $u$, substitute these in the product equation $rs = -24000$
$$(100 - u) (100 + u) = -24000$$
Simplify by expanding $(a -b) (a + b) = a^2 – b^2$
$$10000 - u^2 = -24000$$
Simplify the expression by subtracting $10000$ on both sides
$$-u^2 = -24000-10000 = -34000$$
Simplify the expression by multiplying $-1$ on both sides and take the square root to obtain the value of unknown variable $u$
