Factor the expression by grouping. First, the expression needs to be rewritten as $6b^{2}+pb+qb-24$. To find $p$ and $q$, set up a system to be solved.
$$p+q=7$$ $$pq=6\left(-24\right)=-144$$
Since $pq$ is negative, $p$ and $q$ have the opposite signs. Since $p+q$ is positive, the positive number has greater absolute value than the negative. List all such integer pairs that give product $-144$.
Rewrite $6b^{2}+7b-24$ as $\left(6b^{2}-9b\right)+\left(16b-24\right)$.
$$\left(6b^{2}-9b\right)+\left(16b-24\right)$$
Factor out $3b$ in the first and $8$ in the second group.
$$3b\left(2b-3\right)+8\left(2b-3\right)$$
Factor out common term $2b-3$ by using distributive property.
$$\left(2b-3\right)\left(3b+8\right)$$
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 $ax^{2}+bx+c=0$.
$$6b^{2}+7b-24=0$$
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 $b=\frac{-7±25}{12}$ when $±$ is plus. Add $-7$ to $25$.
$$b=\frac{18}{12}$$
Reduce the fraction $\frac{18}{12}$ to lowest terms by extracting and canceling out $6$.
$$b=\frac{3}{2}$$
Now solve the equation $b=\frac{-7±25}{12}$ when $±$ is minus. Subtract $25$ from $-7$.
$$b=-\frac{32}{12}$$
Reduce the fraction $\frac{-32}{12}$ to lowest terms by extracting and canceling out $4$.
$$b=-\frac{8}{3}$$
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 $-\frac{8}{3}$ for $x_{2}$.
Multiply $\frac{2b-3}{2}$ times $\frac{3b+8}{3}$ 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 $6$
$$x ^ 2 +\frac{7}{6}x -4 = 0$$
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 $rs = C$
$$r + s = -\frac{7}{6} $$ $$ rs = -4$$
Two numbers $r$ and $s$ sum up to $-\frac{7}{6}$ exactly when the average of the two numbers is $\frac{1}{2}*-\frac{7}{6} = -\frac{7}{12}$. 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 $u$.
