Factor the expression by grouping. First, the expression needs to be rewritten as $3x^{2}+ax+bx-6$. To find $a$ and $b$, set up a system to be solved.
$$a+b=-7$$ $$ab=3\left(-6\right)=-18$$
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 $-18$.
$$1,-18$$ $$2,-9$$ $$3,-6$$
Calculate the sum for each pair.
$$1-18=-17$$ $$2-9=-7$$ $$3-6=-3$$
The solution is the pair that gives sum $-7$.
$$a=-9$$ $$b=2$$
Rewrite $3x^{2}-7x-6$ as $\left(3x^{2}-9x\right)+\left(2x-6\right)$.
$$\left(3x^{2}-9x\right)+\left(2x-6\right)$$
Factor out $3x$ in the first and $2$ in the second group.
$$3x\left(x-3\right)+2\left(x-3\right)$$
Factor out common term $x-3$ by using distributive property.
$$\left(x-3\right)\left(3x+2\right)$$
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 $ax^{2}+bx+c=0$.
$$3x^{2}-7x-6=0$$
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 $x=\frac{7±11}{6}$ when $±$ is plus. Add $7$ to $11$.
$$x=\frac{18}{6}$$
Divide $18$ by $6$.
$$x=3$$
Now solve the equation $x=\frac{7±11}{6}$ when $±$ is minus. Subtract $11$ from $7$.
$$x=-\frac{4}{6}$$
Reduce the fraction $\frac{-4}{6}$ to lowest terms by extracting and canceling out $2$.
$$x=-\frac{2}{3}$$
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 $-\frac{2}{3}$ for $x_{2}$.
Cancel out $3$, the greatest common factor in $3$ and $3$.
$$3x^{2}-7x-6=\left(x-3\right)\left(3x+2\right)$$
Steps Using Direct Factoring Method
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 $3$
$$x ^ 2 -\frac{7}{3}x -2 = 0$$
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 $rs = C$
$$r + s = \frac{7}{3} $$ $$ rs = -2$$
Two numbers $r$ and $s$ sum up to $\frac{7}{3}$ exactly when the average of the two numbers is $\frac{1}{2}*\frac{7}{3} = \frac{7}{6}$. 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 $u$.
$$r = \frac{7}{6} - u$$ $$s = \frac{7}{6} + u$$
To solve for unknown quantity $u$, substitute these in the product equation $rs = -2$
$$(\frac{7}{6} - u) (\frac{7}{6} + u) = -2$$
Simplify by expanding $(a -b) (a + b) = a^2 – b^2$
$$\frac{49}{36} - u^2 = -2$$
Simplify the expression by subtracting $\frac{49}{36}$ on both sides
$$-u^2 = -2-\frac{49}{36} = -\frac{121}{36}$$
Simplify the expression by multiplying $-1$ on both sides and take the square root to obtain the value of unknown variable $u$
