Consider $x^{2}+x-12$. Factor the expression by grouping. First, the expression needs to be rewritten as $x^{2}+ax+bx-12$. To find $a$ and $b$, set up a system to be solved.
$$a+b=1$$ $$ab=1\left(-12\right)=-12$$
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 $-12$.
$$-1,12$$ $$-2,6$$ $$-3,4$$
Calculate the sum for each pair.
$$-1+12=11$$ $$-2+6=4$$ $$-3+4=1$$
The solution is the pair that gives sum $1$.
$$a=-3$$ $$b=4$$
Rewrite $x^{2}+x-12$ as $\left(x^{2}-3x\right)+\left(4x-12\right)$.
$$\left(x^{2}-3x\right)+\left(4x-12\right)$$
Factor out $x$ in the first and $4$ in the second group.
$$x\left(x-3\right)+4\left(x-3\right)$$
Factor out common term $x-3$ by using distributive property.
$$\left(x-3\right)\left(x+4\right)$$
Rewrite the complete factored expression.
$$2\left(x-3\right)\left(x+4\right)$$
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 $ax^{2}+bx+c=0$.
$$2x^{2}+2x-24=0$$
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.
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 $2$
$$x ^ 2 +1x -12 = 0$$
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 $rs = C$
$$r + s = -1 $$ $$ rs = -12$$
Two numbers $r$ and $s$ sum up to $-1$ exactly when the average of the two numbers is $\frac{1}{2}*-1 = -\frac{1}{2}$. 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 $u$.
$$r = -\frac{1}{2} - u$$ $$s = -\frac{1}{2} + u$$
To solve for unknown quantity $u$, substitute these in the product equation $rs = -12$
$$(-\frac{1}{2} - u) (-\frac{1}{2} + u) = -12$$
Simplify by expanding $(a -b) (a + b) = a^2 – b^2$
$$\frac{1}{4} - u^2 = -12$$
Simplify the expression by subtracting $\frac{1}{4}$ on both sides
$$-u^2 = -12-\frac{1}{4} = -\frac{49}{4}$$
Simplify the expression by multiplying $-1$ on both sides and take the square root to obtain the value of unknown variable $u$
