Factor the expression by grouping. First, the expression needs to be rewritten as $x^{2}+ax+bx+20$. To find $a$ and $b$, set up a system to be solved.
$$a+b=-9$$ $$ab=1\times 20=20$$
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 $20$.
$$-1,-20$$ $$-2,-10$$ $$-4,-5$$
Calculate the sum for each pair.
$$-1-20=-21$$ $$-2-10=-12$$ $$-4-5=-9$$
The solution is the pair that gives sum $-9$.
$$a=-5$$ $$b=-4$$
Rewrite $x^{2}-9x+20$ as $\left(x^{2}-5x\right)+\left(-4x+20\right)$.
$$\left(x^{2}-5x\right)+\left(-4x+20\right)$$
Factor out $x$ in the first and $-4$ in the second group.
$$x\left(x-5\right)-4\left(x-5\right)$$
Factor out common term $x-5$ by using distributive property.
$$\left(x-5\right)\left(x-4\right)$$
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 $ax^{2}+bx+c=0$.
$$x^{2}-9x+20=0$$
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.
