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 $3$
$$x ^ 2 +\frac{2}{3}x +2 = 0$$
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 $rs = C$
$$r + s = -\frac{2}{3} $$ $$ rs = 2$$
Two numbers $r$ and $s$ sum up to $-\frac{2}{3}$ exactly when the average of the two numbers is $\frac{1}{2}*-\frac{2}{3} = -\frac{1}{3}$. 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 $u$.
$$r = -\frac{1}{3} - u$$ $$s = -\frac{1}{3} + u$$
To solve for unknown quantity $u$, substitute these in the product equation $rs = 2$
$$(-\frac{1}{3} - u) (-\frac{1}{3} + u) = 2$$
Simplify by expanding $(a -b) (a + b) = a^2 – b^2$
$$\frac{1}{9} - u^2 = 2$$
Simplify the expression by subtracting $\frac{1}{9}$ on both sides
$$-u^2 = 2-\frac{1}{9} = \frac{17}{9}$$
Simplify the expression by multiplying $-1$ on both sides and take the square root to obtain the value of unknown variable $u$
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 $nax^{n-1}$.
