Rearrange the polynomial to put it in standard form. Place the terms in order from highest to lowest power.
$$6x^{2}-7x-3$$
Factor the expression by grouping. First, the expression needs to be rewritten as $6x^{2}+ax+bx-3$. To find $a$ and $b$, set up a system to be solved.
$$a+b=-7$$ $$ab=6\left(-3\right)=-18$$
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 $6x^{2}-7x-3$ as $\left(6x^{2}-9x\right)+\left(2x-3\right)$.
$$\left(6x^{2}-9x\right)+\left(2x-3\right)$$
Factor out $3x$ in $6x^{2}-9x$.
$$3x\left(2x-3\right)+2x-3$$
Factor out common term $2x-3$ by using distributive property.
$$\left(2x-3\right)\left(3x+1\right)$$
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 $ax^{2}+bx+c=0$.
$$6x^{2}-7x-3=0$$
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 $x=\frac{7±11}{12}$ when $±$ is plus. Add $7$ to $11$.
$$x=\frac{18}{12}$$
Reduce the fraction $\frac{18}{12}$ to lowest terms by extracting and canceling out $6$.
$$x=\frac{3}{2}$$
Now solve the equation $x=\frac{7±11}{12}$ when $±$ is minus. Subtract $11$ from $7$.
$$x=-\frac{4}{12}$$
Reduce the fraction $\frac{-4}{12}$ to lowest terms by extracting and canceling out $4$.
$$x=-\frac{1}{3}$$
Factor the original expression using $ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right)$. Substitute $\frac{3}{2}$ for $x_{1}$ and $-\frac{1}{3}$ for $x_{2}$.
Multiply $\frac{2x-3}{2}$ times $\frac{3x+1}{3}$ by multiplying numerator times numerator and denominator times denominator. Then reduce the fraction to lowest terms if possible.
