Factor the expression by grouping. First, the expression needs to be rewritten as $y^{2}+ay+by+12$. To find $a$ and $b$, set up a system to be solved.
$$a+b=-7$$ $$ab=1\times 12=12$$
Since $ab$ is positive, $a$ and $b$ have the same sign. Since $a+b$ is negative, $a$ and $b$ are both negative. List all such integer pairs that give product $12$.
$$-1,-12$$ $$-2,-6$$ $$-3,-4$$
Calculate the sum for each pair.
$$-1-12=-13$$ $$-2-6=-8$$ $$-3-4=-7$$
The solution is the pair that gives sum $-7$.
$$a=-4$$ $$b=-3$$
Rewrite $y^{2}-7y+12$ as $\left(y^{2}-4y\right)+\left(-3y+12\right)$.
$$\left(y^{2}-4y\right)+\left(-3y+12\right)$$
Factor out $y$ in the first and $-3$ in the second group.
$$y\left(y-4\right)-3\left(y-4\right)$$
Factor out common term $y-4$ by using distributive property.
$$\left(y-4\right)\left(y-3\right)$$
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 $ax^{2}+bx+c=0$.
$$y^{2}-7y+12=0$$
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 $y=\frac{7±1}{2}$ when $±$ is plus. Add $7$ to $1$.
$$y=\frac{8}{2}$$
Divide $8$ by $2$.
$$y=4$$
Now solve the equation $y=\frac{7±1}{2}$ when $±$ is minus. Subtract $1$ from $7$.
$$y=\frac{6}{2}$$
Divide $6$ by $2$.
$$y=3$$
Factor the original expression using $ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right)$. Substitute $4$ for $x_{1}$ and $3$ for $x_{2}$.
$$y^{2}-7y+12=\left(y-4\right)\left(y-3\right)$$
Steps Using Direct Factoring Method
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 $x^2+Bx+C=0$.
$$x ^ 2 -7x +12 = 0$$
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 $rs = C$
$$r + s = 7 $$ $$ rs = 12$$
Two numbers $r$ and $s$ sum up to $7$ exactly when the average of the two numbers is $\frac{1}{2}*7 = \frac{7}{2}$. 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 $u$.
$$r = \frac{7}{2} - u$$ $$s = \frac{7}{2} + u$$
To solve for unknown quantity $u$, substitute these in the product equation $rs = 12$
$$(\frac{7}{2} - u) (\frac{7}{2} + u) = 12$$
Simplify by expanding $(a -b) (a + b) = a^2 – b^2$
$$\frac{49}{4} - u^2 = 12$$
Simplify the expression by subtracting $\frac{49}{4}$ on both sides
$$-u^2 = 12-\frac{49}{4} = -\frac{1}{4}$$
Simplify the expression by multiplying $-1$ on both sides and take the square root to obtain the value of unknown variable $u$
