Never solve a multi-step problem in the wrong order 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.
P
Parentheses — solve everything inside ( ) first
E
Exponents — handle all powers and roots next
M/D
Multiplication and Division — left to right, equal priority
A/S
Addition and Subtraction — left to right, equal priority
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.
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.
Zero Product Property — the foundation of factoring to solve
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 for solving quadratics — factor first, then apply zero product property.
Inequality Sign-Flip Rule — the one rule students always forget
Never lose points on an inequality problem again
−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].
Four exponent rules that cover almost every problem
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).
Two Methods for Systems — Substitution (plug in) · Elimination (add/subtract rows)
Two reliable methods for solving any linear system
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.
The "Good/Bad Person" trick — never mix up signs again
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 (+×−=−).
|x| = k gives x = k OR x = -k. Always split into TWO equations. Check both answers.
Absolute value means distance from zero — so there are always two cases
Isolate the absolute value first, then split into plus and minus cases
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.
Isolate first
Get |expr| alone before splitting
Two cases
expr = k AND expr = -k — both must be solved
No solution
|x| = negative number is impossible — write No Solution
f(g(x)) means work inside out. g goes first, f goes second. Order matters.
Substitute the entire inner function into the outer function
To find f(g(x)): plug g(x) everywhere you see x in f
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.
Inside out
Evaluate inner function first, plug result into outer
Order matters
f(g(x)) does not equal g(f(x)) in general
Inverse check
Both compositions give x means they are inverse functions
Factor EVERYTHING first. Cancel common factors. State restrictions BEFORE canceling.
The denominator can never be zero — state all restrictions from the original
Factor fully, state restrictions, then cancel — in that order
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.
Factor first
Always factor completely before doing anything else
DMSB — Divide, Multiply, Subtract, Bring down. Repeat until remainder degree is less than divisor.
Dividing polynomials the same way you do long division with numbers
Remainder Theorem: f(r) equals the remainder when dividing by (x minus r)
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.
DMSB
Divide, Multiply, Subtract, Bring down — repeat
Remainder Theorem
f(r) equals the remainder when dividing by (x minus r)
i squared = -1. Powers cycle every 4: i, -1, -i, 1. Divide exponent by 4 and use the remainder.
Imaginary and complex numbers — arithmetic with the square root of negative one
To divide complex numbers, multiply top and bottom by the conjugate of the denominator
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.
Powers of i
Cycle of 4: i, -1, -i, 1. Use remainder of exponent divided by 4.
Multiply
FOIL then replace i squared with -1
Divide
Multiply by conjugate (a minus bi) to get real denominator
Isolate the radical. Raise BOTH sides to the index power. ALWAYS check for extraneous solutions.
Solving equations containing square roots, cube roots, or nth roots
Squaring both sides can introduce false solutions — checking is not optional
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.
Q: What does FOIL stand for and how is it used to multiply binomials?
A: FOIL stands for First · Outer · Inner · Last. To expand (a+b)(c+d): First=a·c, Outer=a·d, Inner=b·c, Last=b·d — then combine like terms. Example: (x+3)(x−2) = x²−2x+3x−6 = x²+x−6. FOIL only works for two binomials — for three or more terms use the distributive property repeatedly.
Q: What does PEMDAS stand for and what is the most common mistake?
A: PEMDAS: Parentheses · Exponents · Multiplication · Division · Addition · Subtraction. Most common mistake: thinking Multiplication always beats Division and Addition always beats Subtraction. Wrong — M/D are equal priority (work left to right), and so are A/S. Example: 10÷2×5 = 25, not 1. Do 10÷2=5 first (leftmost M/D), then 5×5=25.
Q: State the Quadratic Formula and explain what the discriminant tells you.
A: x = (−b ± √(b²−4ac)) / 2a — solves any ax²+bx+c=0. The discriminant is b²−4ac (expression under the square root). If positive → two distinct real roots. If zero → exactly one real root (repeated root). If negative → no real roots (two complex roots). Memory: sing it to "Pop Goes the Weasel."
Q: What are the four basic exponent rules?
A: (1) Multiply same base → ADD exponents: x³·x⁴=x⁷. (2) Divide same base → SUBTRACT: x⁶÷x²=x⁴. (3) Power to a power → MULTIPLY: (x³)⁴=x¹². (4) Zero power=1: 7⁰=1. Bonus: negative exponent means reciprocal (x⁻²=1/x²), fractional exponent means root (x^(1/2)=√x).
Q: What does STAR stand for and how does it help with word problems?
A: STAR: Search · Translate · Answer · Review. Search: read the entire problem, identify what is being asked. Translate: convert English to math (key words: "is"=equals, "more than"=add, "less than"=subtract, "per"=divide). Answer: solve using appropriate algebra. Review: plug answer back in and verify it makes logical sense in the real-world context.