Cheat sheet · Quadratics

1 · Factoring — in this order

1. GCF first. Take out the greatest common factor of every term: $12x^2 + 16x = 4x(3x + 4)$ .

2. Special patterns. $a^2 - b^2 = (a + b)(a - b)$ · $a^2 \pm 2ab + b^2 = (a \pm b)^2$ .

3. $x^2 + bx + c$: two numbers that multiply to $c$ and add to $b$. $x^2 - 9x + 18 = (x - 3)(x - 6)$ .

4. $ax^2 + bx + c$ (ac-method): two numbers that multiply to $ac$ and add to $b$ , split the middle term, group.

$2x^2 - 17x + 21:\ ac = 42,\ -3 + (-14) = -17$
$= 2x^2 - 3x - 14x + 21 = x(2x - 3) - 7(2x - 3) = (2x - 3)(x - 7)$

Signs: $c > 0$ → same signs (as $b$ ); $c < 0$ → different signs.

2 · Solving by factoring

Standard form first: everything on one side, $= 0$ .

Zero Product Property: if $ab = 0$ then $a = 0$ or $b = 0$ . Set each factor $= 0$ .

$6x^2 - 2x = 0 \Rightarrow 2x(3x - 1) = 0 \Rightarrow x = 0,\ \tfrac{1}{3}$ — never divide by $x$ (you lose $x = 0$ ).

$(2x + 3) = 0 \Rightarrow x = -\tfrac{3}{2}$ — the sign flips and you divide by the coefficient.

Perfect square → one root: $(4x - 6)^2 = 0 \Rightarrow x = \tfrac{3}{2}$ .

$x^2 = k$ → $x = \pm\sqrt{k}$ (two roots). Word problems: reject roots that make no sense (negative lengths).

3 · Complex numbers

$i = \sqrt{-1}$ , $i^2 = -1$ . $\sqrt{-n} = i\sqrt{n}$, simplified: $\sqrt{-84} = 2i\sqrt{21}$ .

Convert to $i$ before multiplying: $\sqrt{-25}\cdot\sqrt{-36} = 5i \cdot 6i = 30i^2 = -30$ (not $+30$ ).

Powers of

i

repeat every 4:

i^1 = i

i^2 = -1

i^3 = -i

i^4 = 1

Use the remainder of $n \div 4$ : $i^{17} = i^1 = i$ , $i^{84} = i^0 = 1$ .

$a + bi$: $a$ is the real part, $bi$ the imaginary part. Equal ⇔ real parts equal and imaginary parts equal.

Add/subtract: real with real, imaginary with imaginary. Distribute the minus to both parts.

Multiply: FOIL, then $i^2 = -1$ : $(2 + i)(3 - i) = 6 - 2i + 3i - i^2 = 7 + i$ .

Divide: multiply top and bottom by the conjugate ( $c + di \to c - di$ ): $\dfrac{5}{4 - 3i}\cdot\dfrac{4 + 3i}{4 + 3i} = \dfrac{20 + 15i}{25} = \dfrac{4 + 3i}{5}$ .

4 · Quadratic Formula

$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

1. Standard form, write $a$ , $b$ , $c$ with signs. 2. Discriminant first. 3. Simplify the radical ( $\sqrt{84} = 2\sqrt{21}$ ; negative → $i$ ). 4. Divide every term by any common factor.

$(-10)^2 = +100$ — on the calculator use brackets.

$x^2 + 4x - 17 = 0$ : $x = \dfrac{-4 \pm \sqrt{84}}{2} = \dfrac{-4 \pm 2\sqrt{21}}{2} = -2 \pm \sqrt{21}$ .

5 · The discriminant $b^2 - 4ac$

Discriminant	Roots
$D > 0, \text{perfect square}$	2 real, rational (real and unequal)
$D > 0, \text{not a} \text{perfect square}$	2 real, irrational (real and unequal)
$D = 0$	1 real, rational root (real and equal)
$D < 0$	2 complex roots (imaginary) — not “no solution”

6 · Exact answers (how it's marked)

Calculator is allowed, but answers are exact: fractions and radicals, not decimals.

Radicals simplified, fractions reduced, no $i$ in a denominator, complex answers as $a + bi$ (or $\tfrac{p + qi}{r}$ ).

Show each step: standard form → factor / substitute → simplify → answer.