Factor the expression by grouping. First, the expression needs to be rewritten as $3x^{2}+ax+bx-8$. To find $a$ and $b$, set up a system to be solved.
$$a+b=23$$ $$ab=3\left(-8\right)=-24$$
Since $ab$ is negative, $a$ and $b$ have the opposite signs. Since $a+b$ is positive, the positive number has greater absolute value than the negative. List all such integer pairs that give product $-24$.
$$-1,24$$ $$-2,12$$ $$-3,8$$ $$-4,6$$
Calculate the sum for each pair.
$$-1+24=23$$ $$-2+12=10$$ $$-3+8=5$$ $$-4+6=2$$
The solution is the pair that gives sum $23$.
$$a=-1$$ $$b=24$$
Rewrite $3x^{2}+23x-8$ as $\left(3x^{2}-x\right)+\left(24x-8\right)$.
$$\left(3x^{2}-x\right)+\left(24x-8\right)$$
Factor out $x$ in the first and $8$ in the second group.
$$x\left(3x-1\right)+8\left(3x-1\right)$$
Factor out common term $3x-1$ by using distributive property.
$$\left(3x-1\right)\left(x+8\right)$$
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 $ax^{2}+bx+c=0$.
$$3x^{2}+23x-8=0$$
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 $x=\frac{-23±25}{6}$ when $±$ is plus. Add $-23$ to $25$.
$$x=\frac{2}{6}$$
Reduce the fraction $\frac{2}{6}$ to lowest terms by extracting and canceling out $2$.
$$x=\frac{1}{3}$$
Now solve the equation $x=\frac{-23±25}{6}$ when $±$ is minus. Subtract $25$ from $-23$.
$$x=-\frac{48}{6}$$
Divide $-48$ by $6$.
$$x=-8$$
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 $-8$ 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 $3$
$$x ^ 2 +\frac{23}{3}x -\frac{8}{3} = 0$$
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 $rs = C$
$$r + s = -\frac{23}{3} $$ $$ rs = -\frac{8}{3}$$
Two numbers $r$ and $s$ sum up to $-\frac{23}{3}$ exactly when the average of the two numbers is $\frac{1}{2}*-\frac{23}{3} = -\frac{23}{6}$. 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 $u$.
