Rearrange the polynomial to put it in standard form. Place the terms in order from highest to lowest power.
$$-6x^{2}+7x-2$$
Factor the expression by grouping. First, the expression needs to be rewritten as $-6x^{2}+ax+bx-2$. To find $a$ and $b$, set up a system to be solved.
$$a+b=7$$ $$ab=-6\left(-2\right)=12$$
Since $ab$ is positive, $a$ and $b$ have the same sign. Since $a+b$ is positive, $a$ and $b$ are both positive. 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 $-6x^{2}+7x-2$ as $\left(-6x^{2}+4x\right)+\left(3x-2\right)$.
$$\left(-6x^{2}+4x\right)+\left(3x-2\right)$$
Factor out $2x$ in the first and $-1$ in the second group.
$$2x\left(-3x+2\right)-\left(-3x+2\right)$$
Factor out common term $-3x+2$ by using distributive property.
$$\left(-3x+2\right)\left(2x-1\right)$$
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 $ax^{2}+bx+c=0$.
$$-6x^{2}+7x-2=0$$
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 $x=\frac{-7±1}{-12}$ when $±$ is plus. Add $-7$ to $1$.
$$x=-\frac{6}{-12}$$
Reduce the fraction $\frac{-6}{-12}$ to lowest terms by extracting and canceling out $6$.
$$x=\frac{1}{2}$$
Now solve the equation $x=\frac{-7±1}{-12}$ when $±$ is minus. Subtract $1$ from $-7$.
$$x=-\frac{8}{-12}$$
Reduce the fraction $\frac{-8}{-12}$ to lowest terms by extracting and canceling out $4$.
$$x=\frac{2}{3}$$
Factor the original expression using $ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right)$. Substitute $\frac{1}{2}$ for $x_{1}$ and $\frac{2}{3}$ for $x_{2}$.
Multiply $\frac{-2x+1}{-2}$ times $\frac{-3x+2}{-3}$ by multiplying numerator times numerator and denominator times denominator. Then reduce the fraction to lowest terms if possible.
