Factoring

GCF

• The idea is simply $ab + ac = a(b + c)$

• Ex: $2x + 4 = 2(x + 2)$

• This is because both are divisible by 2.

Difference of Two Perfect Squares (DOTS)

• The idea is $x² + y² = (x+y)(x-y)$

• Ex: $x² - 4 = (x + 2)(x - 2)$

• This is because when factoring, you get $x² + 2x - 2x - 4$ and $2x - 2x = 0$, so we are just left with $x² - 4$.

Trinomial

• The idea is find two numbers that add up to b and multiply to c.

• Ex: $x² + x - 6 = (x+3)(x-2)$

• This is because b is $1$ and c is $-6$ so 2 numbers that would multiply to c and add up to b is $3$ and $-2$. $3 * -2 = -6$, $3 + (-2) = 1$.

AC Method

• The idea is similar to trinomial but this is only if $a ≠ 1$.

• Ex: $2x² + 15x + 18 = (2x + 3)(x + 6)$

• This is because a $*$ c = $36$, and $3$ and $12$ both add up to 15 and multiply to 36. We can then do $2x² + 12x + 3x + 18$, then separate it up into two pairs:

  ($(2x² + 12x) + (3x + 18)$). Then we can use GCF to separate a common factor from both pairs. $2x(x+6) + 3(x+6)$ → $(2x+3)(x+6)$

Quadratic Formula

• Formula is $x = \frac{-b \pm \sqrt{b² - 4ac}}{2a}$

• Ex: $-x² + 4x - 20$

• We can break this down into what $a$, $b$, and $c$ are. $a$ is $-1$, since $x²$ has a coefficient of $-1$, $b$ is $4$, since $4x$ has a coefficient of $4$, $c$ is $-20$, this is common sense.

• Let’s plug this into our equation: $x = \frac{-(4) \pm \sqrt{(4)² - 4(-1)(-20)}}{2(-1)}$ → $x = \frac{-4 \pm \sqrt{16 - 80}}{-2}$ → $x = \frac{-4 \pm \sqrt{-64}}{-2}$ → $x = 2 \pm \frac{\sqrt{-64}}{-2}$ → $x = 2 \pm \frac{\sqrt{(64)(-1)}}{-2}$ →$x = 2 \pm \frac{8i}{-2}$ → $x = 2 \pm -4i$