Factor the expression by grouping. First, the expression needs to be rewritten as $12x^{2}+ax+bx+1$. To find $a$ and $b$, set up a system to be solved.
$$a+b=-7$$ $$ab=12\times 1=12$$
Since $ab$ is positive, $a$ and $b$ have the same sign. Since $a+b$ is negative, $a$ and $b$ are both negative. List all such integer pairs that give product $12$.
$$-1,-12$$ $$-2,-6$$ $$-3,-4$$
Calculate the sum for each pair.
$$-1-12=-13$$ $$-2-6=-8$$ $$-3-4=-7$$
The solution is the pair that gives sum $-7$.
$$a=-4$$ $$b=-3$$
Rewrite $12x^{2}-7x+1$ as $\left(12x^{2}-4x\right)+\left(-3x+1\right)$.
$$\left(12x^{2}-4x\right)+\left(-3x+1\right)$$
Factor out $4x$ in the first and $-1$ in the second group.
$$4x\left(3x-1\right)-\left(3x-1\right)$$
Factor out common term $3x-1$ by using distributive property.
$$\left(3x-1\right)\left(4x-1\right)$$
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 $ax^{2}+bx+c=0$.
$$12x^{2}-7x+1=0$$
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.
Now solve the equation $x=\frac{7±1}{24}$ when $±$ is plus. Add $7$ to $1$.
$$x=\frac{8}{24}$$
Reduce the fraction $\frac{8}{24}$ to lowest terms by extracting and canceling out $8$.
$$x=\frac{1}{3}$$
Now solve the equation $x=\frac{7±1}{24}$ when $±$ is minus. Subtract $1$ from $7$.
$$x=\frac{6}{24}$$
Reduce the fraction $\frac{6}{24}$ to lowest terms by extracting and canceling out $6$.
$$x=\frac{1}{4}$$
Factor the original expression using $ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right)$. Substitute $\frac{1}{3}$ for $x_{1}$ and $\frac{1}{4}$ for $x_{2}$.
Multiply $\frac{3x-1}{3}$ times $\frac{4x-1}{4}$ by multiplying numerator times numerator and denominator times denominator. Then reduce the fraction to lowest terms if possible.
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 $12$
$$x ^ 2 -\frac{7}{12}x +\frac{1}{12} = 0$$
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 $rs = C$
$$r + s = \frac{7}{12} $$ $$ rs = \frac{1}{12}$$
Two numbers $r$ and $s$ sum up to $\frac{7}{12}$ exactly when the average of the two numbers is $\frac{1}{2}*\frac{7}{12} = \frac{7}{24}$. 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 $u$.
$$r = \frac{7}{24} - u$$ $$s = \frac{7}{24} + u$$
To solve for unknown quantity $u$, substitute these in the product equation $rs = \frac{1}{12}$
