Factor the expression by grouping. First, the expression needs to be rewritten as $35y^{2}+ay+by-12$. To find $a$ and $b$, set up a system to be solved.
$$a+b=13$$ $$ab=35\left(-12\right)=-420$$
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 $-420$.
Rewrite $35y^{2}+13y-12$ as $\left(35y^{2}-15y\right)+\left(28y-12\right)$.
$$\left(35y^{2}-15y\right)+\left(28y-12\right)$$
Factor out $5y$ in the first and $4$ in the second group.
$$5y\left(7y-3\right)+4\left(7y-3\right)$$
Factor out common term $7y-3$ by using distributive property.
$$\left(7y-3\right)\left(5y+4\right)$$
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 $ax^{2}+bx+c=0$.
$$35y^{2}+13y-12=0$$
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 $y=\frac{-13±43}{70}$ when $±$ is plus. Add $-13$ to $43$.
$$y=\frac{30}{70}$$
Reduce the fraction $\frac{30}{70}$ to lowest terms by extracting and canceling out $10$.
$$y=\frac{3}{7}$$
Now solve the equation $y=\frac{-13±43}{70}$ when $±$ is minus. Subtract $43$ from $-13$.
$$y=-\frac{56}{70}$$
Reduce the fraction $\frac{-56}{70}$ to lowest terms by extracting and canceling out $14$.
$$y=-\frac{4}{5}$$
Factor the original expression using $ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right)$. Substitute $\frac{3}{7}$ for $x_{1}$ and $-\frac{4}{5}$ for $x_{2}$.
Multiply $\frac{7y-3}{7}$ times $\frac{5y+4}{5}$ 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 $35$
$$x ^ 2 +\frac{13}{35}x -\frac{12}{35} = 0$$
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 $rs = C$
$$r + s = -\frac{13}{35} $$ $$ rs = -\frac{12}{35}$$
Two numbers $r$ and $s$ sum up to $-\frac{13}{35}$ exactly when the average of the two numbers is $\frac{1}{2}*-\frac{13}{35} = -\frac{13}{70}$. 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 $u$.
