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.
$$a+b=7$$ $$ab=6\left(-5\right)=-30$$
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 $-30$.
$$-1,30$$ $$-2,15$$ $$-3,10$$ $$-5,6$$
Calculate the sum for each pair.
$$-1+30=29$$ $$-2+15=13$$ $$-3+10=7$$ $$-5+6=1$$
The solution is the pair that gives sum $7$.
$$a=-3$$ $$b=10$$
Rewrite $6x^{2}+7x-5$ as $\left(6x^{2}-3x\right)+\left(10x-5\right)$.
$$\left(6x^{2}-3x\right)+\left(10x-5\right)$$
Factor out $3x$ in the first and $5$ in the second group.
$$3x\left(2x-1\right)+5\left(2x-1\right)$$
Factor out common term $2x-1$ by using distributive property.
$$\left(2x-1\right)\left(3x+5\right)$$
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 $ax^{2}+bx+c=0$.
$$6x^{2}+7x-5=0$$
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 $x=\frac{-7±13}{12}$ when $±$ is plus. Add $-7$ to $13$.
$$x=\frac{6}{12}$$
Reduce the fraction $\frac{6}{12}$ to lowest terms by extracting and canceling out $6$.
$$x=\frac{1}{2}$$
Now solve the equation $x=\frac{-7±13}{12}$ when $±$ is minus. Subtract $13$ from $-7$.
$$x=-\frac{20}{12}$$
Reduce the fraction $\frac{-20}{12}$ to lowest terms by extracting and canceling out $4$.
$$x=-\frac{5}{3}$$
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 $-\frac{5}{3}$ for $x_{2}$.
Multiply $\frac{2x-1}{2}$ times $\frac{3x+5}{3}$ 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 $6$
$$x ^ 2 +\frac{7}{6}x -\frac{5}{6} = 0$$
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 $rs = C$
$$r + s = -\frac{7}{6} $$ $$ rs = -\frac{5}{6}$$
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 $u$.
