Algebra Memory Tricks | MemoryTricks

🔢 Algebra

FOIL

Multiplying Two Binomials

Expand (a+b)(c+d) every time without mistakes

First, Outer, Inner, Last. Every binomial multiplication follows this exact order. Never miss a term again.

F

First — multiply the first terms of each binomial

O

Outer — multiply the outermost terms

I

Inner — multiply the innermost terms

L

Last — multiply the last terms of each binomial

🔢 Algebra · Order of Operations

Please Excuse My Dear Aunt Sally

PEMDAS — The US Standard (Used in all American schools)

Never solve a multi-step problem in the wrong order again

When a math problem has multiple operations, you MUST solve them in a specific order — or you'll get the wrong answer every time. In the United States, that order is PEMDAS.

P

Parentheses — solve everything inside ( ) first

E

Exponents — handle all powers and roots next

M

Multiplication — left to right

D

Division — left to right (same level as Multiplication)

A

Addition — left to right

S

Subtraction — left to right (same level as Addition)

⚠️ The #1 Mistake Students Make

Most students think Multiplication always beats Division, and Addition always beats Subtraction. That's wrong. Once you reach M/D or A/S — just work left to right, whichever comes first wins.

📝 Quick Example

Solve: 3 + 6 × (5 + 4) ÷ 3 − 7
Step 1 — Parentheses: 5+4 = 9 → 3 + 6 × 9 ÷ 3 − 7
Step 2 — No exponents
Step 3 — Multiply/Divide left to right: 6×9 = 54 → 3 + 54 ÷ 3 − 7
Step 4 — Continue: 54÷3 = 18 → 3 + 18 − 7
Step 5 — Add/Subtract left to right: 3+18 = 21 → 21−7 = 14 ✓

🔢 Algebra

"Difference of Squares"

a² − b² = (a+b)(a−b)

Factor any difference of two perfect squares instantly

Spot two perfect squares being subtracted? Factor immediately: a² − b² = (a+b)(a−b). This pattern is on every algebra exam.

🔢 Algebra

Slope = Rise / Run

Slope Formula

Never confuse rise and run again

Rise is the vertical change (y₂ − y₁). Run is the horizontal change (x₂ − x₁). Rise over Run = slope. Think: you rise UP before you run ACROSS.

🔢 Algebra

"Slide and Divide"

Factoring ax² + bx + c

Factor trinomials with a leading coefficient fast

Multiply a and c together (slide). Factor that product with b. Divide by a and simplify. Works every time for hard trinomials.

Quadratic Formula

Quadratic formula: x = (-b ± √(b²-4ac)) / 2a — 'Pop goes the weasel' rhythm helps

Quadratic Formula

Solve any quadratic ax²+bx+c=0 — memorize this cold

Sing it to 'Pop Goes the Weasel': 'x equals negative b, plus or minus square root, b squared minus 4ac, all over 2a.' The discriminant (b²-4ac): positive = 2 real roots, zero = 1 real root, negative = no real roots.

Zero Product Property

Zero product property: if (x-3)(x+2)=0, then x=3 or x=-2

Zero Product Property

If a product equals zero, at least one factor must be zero

Set each factor equal to zero and solve. (x-5)(x+4)=0 → x=5 or x=-4. This is why factoring works to solve quadratics. Factor the quadratic, set each factor to zero, solve for x.

Solving Systems of Equations

Systems of equations: substitution (plug in) or elimination (add/subtract rows)

Solving Systems of Equations

Two methods for finding where two equations intersect

Substitution: isolate one variable, plug into other equation. Best when one equation is already solved for a variable. Elimination: multiply equations to make coefficients match, add or subtract to eliminate one variable. Best when coefficients are easy to match.

Exponent Rules

Exponent rules: xᵃ · xᵇ = xᵃ⁺ᵇ. xᵃ ÷ xᵇ = xᵃ⁻ᵇ. (xᵃ)ᵇ = xᵃᵇ. x⁰ = 1.

Exponent Rules

Four rules that cover almost every exponent problem

Multiply same base: ADD exponents. Divide same base: SUBTRACT exponents. Power to a power: MULTIPLY exponents. Anything to the zero power = 1. Negative exponent: x⁻ⁿ = 1/xⁿ (flip to denominator). Fractional exponent: x^(1/n) = ⁿ√x.

Multiply

xᵃ · xᵇ = xᵃ⁺ᵇ — add exponents

Divide

xᵃ ÷ xᵇ = xᵃ⁻ᵇ — subtract exponents

Power

(xᵃ)ᵇ = xᵃᵇ — multiply exponents

Zero

x⁰ = 1 always

Absolute Value

Absolute value: |x| = distance from zero. Always positive. |x| = a means x = a or x = -a

Absolute Value

Distance from zero — always non-negative

|-5| = 5. |3| = 3. When solving |x-2| = 5: set up two equations, x-2 = 5 (x=7) AND x-2 = -5 (x=-3). For inequalities: |x| < a means -a < x < a (AND). |x| > a means x > a OR x < -a.

Function Notation

Function notation: f(x) means 'f of x' — plug x in wherever you see the variable

Function Notation

Reading and evaluating function notation

f(x) = 2x + 3. Find f(4): replace x with 4 → f(4) = 2(4)+3 = 11. f(x+1): replace x with (x+1) → 2(x+1)+3 = 2x+5. Composition: f(g(x)) means find g(x) first, then plug that into f.

Inequality Rules

Inequalities: flip the sign when multiplying or dividing by a NEGATIVE number

Inequality Rules

The one rule students always forget — flip when dividing by negative

-2x > 6 → divide both sides by -2 → x < -3 (sign flips). On a number line: < and > use open circles. ≤ and ≥ use closed circles. Interval notation: x > 3 is (3, ∞). x ≤ 5 is (-∞, 5].

🔢 Algebra

STAR

Solving Word Problems

Never get lost in a word problem again

Word problems feel overwhelming because students jump straight to solving. STAR slows you down and keeps you on track every time.

S

Search — read the problem carefully, identify what's being asked

T

Translate — turn the words into a math equation

A

Answer — solve the equation

R

Review — check that your answer makes sense in context

🔢 Algebra

Good × Bad = Bad. Bad × Bad = Good.

Multiplying Positive & Negative Numbers

The "Good Person / Bad Person" trick — never mix up signs again

Think of positive numbers as good people and negative numbers as bad people. A good thing happening to a good person = good. A bad thing happening to a bad person = also good. This logic mirrors the sign rules exactly.

+ × +

A good thing happening to a good person = Good (+)

+ × −

A good thing happening to a bad person = Bad (−)

− × +

A bad thing happening to a good person = Bad (−)

− × −

A bad thing happening to a bad person = Good (+)

🔢 Algebra

y = mx + b → "Make Believe"

Slope-Intercept Form

Remember which is slope and which is the y-intercept — instantly

Students constantly confuse m and b in y = mx + b. The trick: M is the Move (slope — how steep the line moves). B is where the line Begins (the y-intercept — where it crosses the y-axis). Start at b, then move by m.

m

Move — the slope (rise over run). Positive = line goes up. Negative = line goes down.

b

Beginning — the y-intercept. Where the line crosses the y-axis (when x = 0).

Example: y = 3x + 2 → slope is 3 (moves up 3, right 1), y-intercept is 2 (starts at point (0, 2)).

🔢 Algebra

PIES

Solving Word Problems (Visual Method)

A visual 4-step approach to tackling any word problem

PIES is especially useful when STAR isn't enough — it forces you to draw before you solve, which catches mistakes before they happen.

P

Picture — draw a simple sketch based on the problem

I

Information — circle key numbers and words, write them next to your sketch

E

Equation — write an equation that fits the information

S

Solve — solve the equation and check your answer

🔢 Algebra

"The Denominator is always Down"

Numerator vs. Denominator

Never mix up the top and bottom of a fraction again

In any fraction, students constantly swap numerator and denominator under pressure. The fix is simple: Denominator = Down. It's always the bottom number. The numerator is on top — think of it as Numerator = North (up).

N

Numerator = North — always the top number of a fraction

D

Denominator = Down — always the bottom number of a fraction

Example: In ¾ — the numerator is 3 (top/North), the denominator is 4 (bottom/Down).

Algebra tricks that make equations click

Memory tricks