Factor the expression by grouping. First, the expression needs to be rewritten as $11x^{2}+ax+bx-9$. To find $a$ and $b$, set up a system to be solved.
$$a+b=-30$$ $$ab=11\left(-9\right)=-99$$
Since $ab$ is negative, $a$ and $b$ have the opposite signs. Since $a+b$ is negative, the negative number has greater absolute value than the positive. List all such integer pairs that give product $-99$.
$$1,-99$$ $$3,-33$$ $$9,-11$$
Calculate the sum for each pair.
$$1-99=-98$$ $$3-33=-30$$ $$9-11=-2$$
The solution is the pair that gives sum $-30$.
$$a=-33$$ $$b=3$$
Rewrite $11x^{2}-30x-9$ as $\left(11x^{2}-33x\right)+\left(3x-9\right)$.
$$\left(11x^{2}-33x\right)+\left(3x-9\right)$$
Factor out $11x$ in the first and $3$ in the second group.
$$11x\left(x-3\right)+3\left(x-3\right)$$
Factor out common term $x-3$ by using distributive property.
$$\left(x-3\right)\left(11x+3\right)$$
Steps Using the Quadratic Formula
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 $ax^{2}+bx+c=0$.
$$11x^{2}-30x-9=0$$
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 $x=\frac{30±36}{22}$ when $±$ is plus. Add $30$ to $36$.
$$x=\frac{66}{22}$$
Divide $66$ by $22$.
$$x=3$$
Now solve the equation $x=\frac{30±36}{22}$ when $±$ is minus. Subtract $36$ from $30$.
$$x=-\frac{6}{22}$$
Reduce the fraction $\frac{-6}{22}$ to lowest terms by extracting and canceling out $2$.
$$x=-\frac{3}{11}$$
Factor the original expression using $ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right)$. Substitute $3$ for $x_{1}$ and $-\frac{3}{11}$ 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 $11$
$$x ^ 2 -\frac{30}{11}x -\frac{9}{11} = 0$$
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 $rs = C$
$$r + s = \frac{30}{11} $$ $$ rs = -\frac{9}{11}$$
Two numbers $r$ and $s$ sum up to $\frac{30}{11}$ exactly when the average of the two numbers is $\frac{1}{2}*\frac{30}{11} = \frac{15}{11}$. 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 $u$.
