Factor the expression by grouping. First, the expression needs to be rewritten as $m^{2}+am+bm-32$. To find $a$ and $b$, set up a system to be solved.
$$a+b=-14$$ $$ab=1\left(-32\right)=-32$$
Since $ab$ is negative, $a$ and $b$ have the opposite signs. Since $a+b$ is negative, the negative number has greater absolute value than the positive. List all such integer pairs that give product $-32$.
$$1,-32$$ $$2,-16$$ $$4,-8$$
Calculate the sum for each pair.
$$1-32=-31$$ $$2-16=-14$$ $$4-8=-4$$
The solution is the pair that gives sum $-14$.
$$a=-16$$ $$b=2$$
Rewrite $m^{2}-14m-32$ as $\left(m^{2}-16m\right)+\left(2m-32\right)$.
$$\left(m^{2}-16m\right)+\left(2m-32\right)$$
Factor out $m$ in the first and $2$ in the second group.
$$m\left(m-16\right)+2\left(m-16\right)$$
Factor out common term $m-16$ by using distributive property.
$$\left(m-16\right)\left(m+2\right)$$
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 $ax^{2}+bx+c=0$.
$$m^{2}-14m-32=0$$
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 $x^2+Bx+C=0$.
$$x ^ 2 -14x -32 = 0$$
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 $rs = C$
$$r + s = 14 $$ $$ rs = -32$$
Two numbers $r$ and $s$ sum up to $14$ exactly when the average of the two numbers is $\frac{1}{2}*14 = 7$. 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 $u$.
$$r = 7 - u$$ $$s = 7 + u$$
To solve for unknown quantity $u$, substitute these in the product equation $rs = -32$
$$(7 - u) (7 + u) = -32$$
Simplify by expanding $(a -b) (a + b) = a^2 – b^2$
$$49 - u^2 = -32$$
Simplify the expression by subtracting $49$ on both sides
$$-u^2 = -32-49 = -81$$
Simplify the expression by multiplying $-1$ on both sides and take the square root to obtain the value of unknown variable $u$
$$u^2 = 81$$ $$u = \pm\sqrt{81} = \pm 9 $$
The factors $r$ and $s$ are the solutions to the quadratic equation. Substitute the value of $u$ to compute the $r$ and $s$.
