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 $ax^{2}+bx+c=0$.
$$2x^{2}-8x+5=0$$
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 $x=\frac{8±2\sqrt{6}}{4}$ when $±$ is plus. Add $8$ to $2\sqrt{6}$.
$$x=\frac{2\sqrt{6}+8}{4}$$
Divide $8+2\sqrt{6}$ by $4$.
$$x=\frac{\sqrt{6}}{2}+2$$
Now solve the equation $x=\frac{8±2\sqrt{6}}{4}$ when $±$ is minus. Subtract $2\sqrt{6}$ from $8$.
$$x=\frac{8-2\sqrt{6}}{4}$$
Divide $8-2\sqrt{6}$ by $4$.
$$x=-\frac{\sqrt{6}}{2}+2$$
Factor the original expression using $ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right)$. Substitute $2+\frac{\sqrt{6}}{2}$ for $x_{1}$ and $2-\frac{\sqrt{6}}{2}$ 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 $2$
$$x ^ 2 -4x +\frac{5}{2} = 0$$
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 $rs = C$
$$r + s = 4 $$ $$ rs = \frac{5}{2}$$
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 $rs = \frac{5}{2}$
$$(2 - u) (2 + u) = \frac{5}{2}$$
Simplify by expanding $(a -b) (a + b) = a^2 – b^2$
$$4 - u^2 = \frac{5}{2}$$
Simplify the expression by subtracting $4$ on both sides
$$-u^2 = \frac{5}{2}-4 = -\frac{3}{2}$$
Simplify the expression by multiplying $-1$ on both sides and take the square root to obtain the value of unknown variable $u$
