Factor the expression by grouping. First, the expression needs to be rewritten as $x^{2}+ax+bx+65$. To find $a$ and $b$, set up a system to be solved.
$$a+b=-18$$ $$ab=1\times 65=65$$
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 $65$.
$$-1,-65$$ $$-5,-13$$
Calculate the sum for each pair.
$$-1-65=-66$$ $$-5-13=-18$$
The solution is the pair that gives sum $-18$.
$$a=-13$$ $$b=-5$$
Rewrite $x^{2}-18x+65$ as $\left(x^{2}-13x\right)+\left(-5x+65\right)$.
$$\left(x^{2}-13x\right)+\left(-5x+65\right)$$
Factor out $x$ in the first and $-5$ in the second group.
$$x\left(x-13\right)-5\left(x-13\right)$$
Factor out common term $x-13$ by using distributive property.
$$\left(x-13\right)\left(x-5\right)$$
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 $ax^{2}+bx+c=0$.
$$x^{2}-18x+65=0$$
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.
