Factor the expression by grouping. First, the expression needs to be rewritten as $a^{2}+pa+qa-255$. To find $p$ and $q$, set up a system to be solved.
$$p+q=-2$$ $$pq=1\left(-255\right)=-255$$
Since $pq$ is negative, $p$ and $q$ have the opposite signs. Since $p+q$ is negative, the negative number has greater absolute value than the positive. List all such integer pairs that give product $-255$.
Rewrite $a^{2}-2a-255$ as $\left(a^{2}-17a\right)+\left(15a-255\right)$.
$$\left(a^{2}-17a\right)+\left(15a-255\right)$$
Factor out $a$ in the first and $15$ in the second group.
$$a\left(a-17\right)+15\left(a-17\right)$$
Factor out common term $a-17$ by using distributive property.
$$\left(a-17\right)\left(a+15\right)$$
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 $ax^{2}+bx+c=0$.
$$a^{2}-2a-255=0$$
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 $x^2+Bx+C=0$.
$$x ^ 2 -2x -255 = 0$$
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 $rs = C$
$$r + s = 2 $$ $$ rs = -255$$
Two numbers $r$ and $s$ sum up to $2$ exactly when the average of the two numbers is $\frac{1}{2}*2 = 1$. 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 $u$.
$$r = 1 - u$$ $$s = 1 + u$$
To solve for unknown quantity $u$, substitute these in the product equation $rs = -255$
$$(1 - u) (1 + u) = -255$$
Simplify by expanding $(a -b) (a + b) = a^2 – b^2$
$$1 - u^2 = -255$$
Simplify the expression by subtracting $1$ on both sides
$$-u^2 = -255-1 = -256$$
Simplify the expression by multiplying $-1$ on both sides and take the square root to obtain the value of unknown variable $u$
$$u^2 = 256$$ $$u = \pm\sqrt{256} = \pm 16 $$
The factors $r$ and $s$ are the solutions to the quadratic equation. Substitute the value of $u$ to compute the $r$ and $s$.
