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 $ax^{2}+bx+c=0$.
$$x^{2}+4x+2=0$$
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.
$$x=\frac{-4±\sqrt{4^{2}-4\times 2}}{2}$$
Square $4$.
$$x=\frac{-4±\sqrt{16-4\times 2}}{2}$$
Multiply $-4$ times $2$.
$$x=\frac{-4±\sqrt{16-8}}{2}$$
Add $16$ to $-8$.
$$x=\frac{-4±\sqrt{8}}{2}$$
Take the square root of $8$.
$$x=\frac{-4±2\sqrt{2}}{2}$$
Now solve the equation $x=\frac{-4±2\sqrt{2}}{2}$ when $±$ is plus. Add $-4$ to $2\sqrt{2}$.
$$x=\frac{2\sqrt{2}-4}{2}$$
Divide $-4+2\sqrt{2}$ by $2$.
$$x=\sqrt{2}-2$$
Now solve the equation $x=\frac{-4±2\sqrt{2}}{2}$ when $±$ is minus. Subtract $2\sqrt{2}$ from $-4$.
$$x=\frac{-2\sqrt{2}-4}{2}$$
Divide $-4-2\sqrt{2}$ by $2$.
$$x=-\sqrt{2}-2$$
Factor the original expression using $ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right)$. Substitute $-2+\sqrt{2}$ for $x_{1}$ and $-2-\sqrt{2}$ 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 $x^2+Bx+C=0$.
$$x ^ 2 +4x +2 = 0$$
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 $rs = C$
$$r + s = -4 $$ $$ rs = 2$$
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 $rs = 2$
$$(-2 - u) (-2 + u) = 2$$
Simplify by expanding $(a -b) (a + b) = a^2 – b^2$
$$4 - u^2 = 2$$
Simplify the expression by subtracting $4$ on both sides
$$-u^2 = 2-4 = -2$$
Simplify the expression by multiplying $-1$ on both sides and take the square root to obtain the value of unknown variable $u$
$$u^2 = 2$$ $$u = \pm\sqrt{2} = \pm \sqrt{2} $$
The factors $r$ and $s$ are the solutions to the quadratic equation. Substitute the value of $u$ to compute the $r$ and $s$.
