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 $ax^{2}+bx+c=0$.
$$x^{2}+3x-1=0$$
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.
$$x=\frac{-3±\sqrt{3^{2}-4\left(-1\right)}}{2}$$
Square $3$.
$$x=\frac{-3±\sqrt{9-4\left(-1\right)}}{2}$$
Multiply $-4$ times $-1$.
$$x=\frac{-3±\sqrt{9+4}}{2}$$
Add $9$ to $4$.
$$x=\frac{-3±\sqrt{13}}{2}$$
Now solve the equation $x=\frac{-3±\sqrt{13}}{2}$ when $±$ is plus. Add $-3$ to $\sqrt{13}$.
$$x=\frac{\sqrt{13}-3}{2}$$
Now solve the equation $x=\frac{-3±\sqrt{13}}{2}$ when $±$ is minus. Subtract $\sqrt{13}$ from $-3$.
$$x=\frac{-\sqrt{13}-3}{2}$$
Factor the original expression using $ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right)$. Substitute $\frac{-3+\sqrt{13}}{2}$ for $x_{1}$ and $\frac{-3-\sqrt{13}}{2}$ 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 $x^2+Bx+C=0$.
$$x ^ 2 +3x -1 = 0$$
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 $rs = C$
$$r + s = -3 $$ $$ rs = -1$$
Two numbers $r$ and $s$ sum up to $-3$ exactly when the average of the two numbers is $\frac{1}{2}*-3 = -\frac{3}{2}$. 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 $u$.
$$r = -\frac{3}{2} - u$$ $$s = -\frac{3}{2} + u$$
To solve for unknown quantity $u$, substitute these in the product equation $rs = -1$
$$(-\frac{3}{2} - u) (-\frac{3}{2} + u) = -1$$
Simplify by expanding $(a -b) (a + b) = a^2 – b^2$
$$\frac{9}{4} - u^2 = -1$$
Simplify the expression by subtracting $\frac{9}{4}$ on both sides
$$-u^2 = -1-\frac{9}{4} = -\frac{13}{4}$$
Simplify the expression by multiplying $-1$ on both sides and take the square root to obtain the value of unknown variable $u$
