Memory tricks for algebra

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

When a math problem has multiple operations, solve them in PEMDAS order. The most common mistake: M/D and A/S are equal priority — work left to right when you reach those levels.

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

Rise is the vertical change (y₂−y₁). Run is the horizontal change (x₂−x₁). Think: you rise UP before you run ACROSS. Positive slope goes up left-to-right; negative slope goes down.

Spot two perfect squares being subtracted? Factor immediately: a²−b²=(a+b)(a−b). Example: x²−9=(x+3)(x−3). This pattern appears on every algebra exam.

Multiply a and c together (slide). Factor that product with b. Divide by a and simplify. Works every time for trinomials where the leading coefficient isn't 1.

Sing to "Pop Goes the Weasel": "x equals negative b, plus or minus square root, b squared minus 4ac, all over 2a." Discriminant b²−4ac: positive=2 real roots, zero=1 root, negative=no real roots.

Set each factor equal to zero and solve. (x−5)(x+4)=0 → x=5 or x=−4. This is why factoring works for solving quadratics — factor first, then apply zero product property.

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

Multiply same base → ADD exponents (x³·x⁴=x⁷). Divide same base → SUBTRACT (x⁶÷x²=x⁴). Power to a power → MULTIPLY (x³)⁴=x¹². Anything to ZERO = 1 (7⁰=1).

Substitution: isolate one variable, plug into the other equation — best when one variable is already isolated. Elimination: multiply equations to match coefficients, then add/subtract to cancel one variable.

Positive = good person, negative = bad person. Good thing to a good person = good (+×+=+). Bad thing to a bad person = also good (−×−=+). Good thing to a bad person = bad (+×−=−).

Step 1: Isolate |expression|. Step 2: Split into expr = +k AND expr = -k. Step 3: Solve both. Step 4: Check for extraneous solutions. Special: |x| = 0 has one solution. |x| = negative has NO solution. |x| less than k means -k less than x less than k. |x| greater than k means x less than -k OR x greater than k.

f(g(x)): substitute g(x) into f. Example: f(x) = x squared + 1, g(x) = 3x gives f(g(x)) = 9x squared + 1. Domain: x must be in domain of g AND g(x) must be in domain of f. Inverse check: f(g(x)) = g(f(x)) = x means they are inverses. Finding inverse: swap x and y, solve for y.

Simplify: factor fully, cancel common factors, state all restrictions. Multiply: factor all, cancel across, multiply remaining. Divide: Keep-Change-Flip then multiply. Add/Subtract: find LCD, convert, combine numerators, simplify. Restriction rule: set every original denominator not equal to zero before canceling anything.

Steps: Divide leading terms, multiply result by divisor, subtract, bring down. Repeat. Synthetic division shortcut works when dividing by (x minus r). Remainder Theorem: remainder when f(x) divided by (x minus r) equals f(r). Factor Theorem: (x minus r) is a factor if and only if f(r) equals zero.

Powers of i cycle: i to the 1 = i, squared = -1, cubed = -i, fourth = 1. For i to the n: divide n by 4 and use remainder (0 gives 1, 1 gives i, 2 gives -1, 3 gives -i). Add/Subtract: combine real and imaginary parts separately. Multiply: FOIL then replace i squared with -1. Divide: multiply by conjugate (a minus bi) to eliminate i from denominator.

Steps: Isolate the radical. Raise both sides to the power matching the index. Solve the resulting equation. CHECK every answer in the original. Two radicals: isolate one, raise to power, isolate second, raise to power again. Even root of a negative has no real solution.