Factor the expression by grouping. First, the expression needs to be rewritten as $6x^{2}+ax+bx+5$. To find $a$ and $b$, set up a system to be solved.
$$a+b=17$$ $$ab=6\times 5=30$$
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 $30$.
$$1,30$$ $$2,15$$ $$3,10$$ $$5,6$$
Calculate the sum for each pair.
$$1+30=31$$ $$2+15=17$$ $$3+10=13$$ $$5+6=11$$
The solution is the pair that gives sum $17$.
$$a=2$$ $$b=15$$
Rewrite $6x^{2}+17x+5$ as $\left(6x^{2}+2x\right)+\left(15x+5\right)$.
$$\left(6x^{2}+2x\right)+\left(15x+5\right)$$
Factor out $2x$ in the first and $5$ in the second group.
$$2x\left(3x+1\right)+5\left(3x+1\right)$$
Factor out common term $3x+1$ by using distributive property.
$$\left(3x+1\right)\left(2x+5\right)$$
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 $ax^{2}+bx+c=0$.
$$6x^{2}+17x+5=0$$
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 $x=\frac{-17±13}{12}$ when $±$ is plus. Add $-17$ to $13$.
$$x=-\frac{4}{12}$$
Reduce the fraction $\frac{-4}{12}$ to lowest terms by extracting and canceling out $4$.
$$x=-\frac{1}{3}$$
Now solve the equation $x=\frac{-17±13}{12}$ when $±$ is minus. Subtract $13$ from $-17$.
$$x=-\frac{30}{12}$$
Reduce the fraction $\frac{-30}{12}$ to lowest terms by extracting and canceling out $6$.
$$x=-\frac{5}{2}$$
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 $-\frac{5}{2}$ for $x_{2}$.
Multiply $\frac{3x+1}{3}$ times $\frac{2x+5}{2}$ 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 $6$
$$x ^ 2 +\frac{17}{6}x +\frac{5}{6} = 0$$
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 $rs = C$
$$r + s = -\frac{17}{6} $$ $$ rs = \frac{5}{6}$$
Two numbers $r$ and $s$ sum up to $-\frac{17}{6}$ exactly when the average of the two numbers is $\frac{1}{2}*-\frac{17}{6} = -\frac{17}{12}$. 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 $u$.
