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 $4$
$$x ^ 2 -\frac{5}{4}x +\frac{3}{2} = 0$$
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 $rs = C$
$$r + s = \frac{5}{4} $$ $$ rs = \frac{3}{2}$$
Two numbers $r$ and $s$ sum up to $\frac{5}{4}$ exactly when the average of the two numbers is $\frac{1}{2}*\frac{5}{4} = \frac{5}{8}$. 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 $u$.
$$r = \frac{5}{8} - u$$ $$s = \frac{5}{8} + u$$
To solve for unknown quantity $u$, substitute these in the product equation $rs = \frac{3}{2}$
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 $nax^{n-1}$.
