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 $x^2+Bx+C=0$.
$$x ^ 2 +4x +5 = 0$$
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 $rs = C$
$$r + s = -4 $$ $$ rs = 5$$
Two numbers $r$ and $s$ sum up to $-4$ exactly when the average of the two numbers is $\frac{1}{2}*-4 = -2$. 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 $u$.
$$r = -2 - u$$ $$s = -2 + u$$
To solve for unknown quantity $u$, substitute these in the product equation $rs = 5$
$$(-2 - u) (-2 + u) = 5$$
Simplify by expanding $(a -b) (a + b) = a^2 – b^2$
$$4 - u^2 = 5$$
Simplify the expression by subtracting $4$ on both sides
$$-u^2 = 5-4 = 1$$
Simplify the expression by multiplying $-1$ on both sides and take the square root to obtain the value of unknown variable $u$
$$u^2 = -1$$ $$u = \pm\sqrt{-1} = \pm i $$
The factors $r$ and $s$ are the solutions to the quadratic equation. Substitute the value of $u$ to compute the $r$ and $s$.
The derivative of a polynomial is the sum of the derivatives of its terms. The derivative of a constant term is $0$. The derivative of $ax^{n}$ is $nax^{n-1}$.
