Factor the expression by grouping. First, the expression needs to be rewritten as $x^{2}+ax+bx+21$. To find $a$ and $b$, set up a system to be solved.
$$a+b=-10$$ $$ab=1\times 21=21$$
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 $21$.
$$-1,-21$$ $$-3,-7$$
Calculate the sum for each pair.
$$-1-21=-22$$ $$-3-7=-10$$
The solution is the pair that gives sum $-10$.
$$a=-7$$ $$b=-3$$
Rewrite $x^{2}-10x+21$ as $\left(x^{2}-7x\right)+\left(-3x+21\right)$.
$$\left(x^{2}-7x\right)+\left(-3x+21\right)$$
Factor out $x$ in the first and $-3$ in the second group.
$$x\left(x-7\right)-3\left(x-7\right)$$
Factor out common term $x-7$ by using distributive property.
$$\left(x-7\right)\left(x-3\right)$$
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 $ax^{2}+bx+c=0$.
$$x^{2}-10x+21=0$$
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.
