SAT Advanced Math Formula Sheet
Quadratics, polynomials, exponents, logarithms, radicals, and rational expressions — all the formulas tested in the Digital SAT Advanced Math domain.
Below are the 18 formulas tested in this category, with the situation each one applies to. Drill until you can write the whole sheet from memory in under two minutes — that's the standard top scorers reach by test day.
Each formula links back to the topic page where it's worked into example problems. If a formula doesn't yet feel automatic, click into the topic page and grind that topic's six practice questions.
Formulas
| Concept | Formula | When to use it |
|---|---|---|
| Quadratic formula | x = (−b ± √(b² − 4ac)) / 2a | Solves any ax² + bx + c = 0. |
| Discriminant | Δ = b² − 4ac | Δ>0 ⇒ 2 real, Δ=0 ⇒ 1, Δ<0 ⇒ 0. |
| Vertex form (parabola) | y = a(x − h)² + k | Vertex at (h, k); axis x = h. |
| Sum/product of roots | r₁ + r₂ = −b/a; r₁r₂ = c/a | Quick checks for factored quadratics. |
| Difference of squares | a² − b² = (a − b)(a + b) | Factor pattern. |
| Perfect-square trinomial | (a ± b)² = a² ± 2ab + b² | Expand or recognize. |
| Exponent product | xᵃ · xᵇ = xᵃ⁺ᵇ | Same base. |
| Exponent quotient | xᵃ / xᵇ = xᵃ⁻ᵇ | Same base. |
| Power of a power | (xᵃ)ᵇ = xᵃᵇ | Multiply exponents. |
| Negative exponent | x⁻ᵃ = 1 / xᵃ | Move base across the bar. |
| Fractional exponent | x^(a/b) = ᵇ√(xᵃ) | Numerator = power, denominator = root. |
| Logarithm definition | log_b(x) = y ⇔ bʸ = x | Translate between forms. |
| Log product rule | log_b(MN) = log_b M + log_b N | Combining logs. |
| Log quotient rule | log_b(M/N) = log_b M − log_b N | Subtraction inside ⇒ division. |
| Log power rule | log_b(Mᵏ) = k·log_b M | Bring exponents in front. |
| Exponential growth/decay | A = P(1 + r)ᵗ ; A = Pe^(rt) | Compounded periodically vs continuously. |
| Function composition | (f ∘ g)(x) = f(g(x)) | Apply g first, then f. |
| Inverse function check | f(f⁻¹(x)) = x and f⁻¹(f(x)) = x | Confirms inverse. |
How to memorise this sheet in a week
Day 1. Read every row out loud. Connect each formula to a worked example by clicking into the topic page on which it appears. Goal: understand why each formula is true, not just memorise it.
Day 2. Cover the right-hand columns and quiz yourself on each concept. Write the formula and the use case from memory. Mark the ones you missed.
Day 3. Re-quiz only the ones you missed yesterday. Keep cycling until your error count hits zero.
Day 4. Re-do the full quiz. You should know every row.
Day 5. Apply each formula in a problem context — pick a topic page from the list below and solve all six questions.
Day 6–7. Sit a real past paper. Note any formulas you blanked on under time pressure and add them back to your daily quiz.
Topics that use these formulas
- Quadratic Equations6 Qs
- Quadratic Functions6 Qs
- Factoring Polynomials6 Qs
- Polynomial Arithmetic6 Qs
- Polynomial Functions6 Qs
- Exponential Functions6 Qs
- Exponential Growth and Decay6 Qs
- Logarithms6 Qs
- Radical Equations6 Qs
- Rational Expressions6 Qs
- Rational Equations6 Qs
- Complex Numbers6 Qs
- Function Notation6 Qs
- Function Transformations6 Qs
- Composite and Inverse Functions6 Qs
Common mistakes when applying these formulas
Three errors account for the majority of dropped points on formula-driven questions:
- Substituting before simplifying. Always simplify symbolically as far as you can before plugging numbers in. It cuts arithmetic errors in half.
- Mixing units. If a formula expects metres and you have centimetres, convert first, not at the end. The number you compute is meaningless until the units agree.
- Forgetting the ½ in area or volume formulas. Triangle, trapezoid, sector, cone, pyramid — all have a ½ or ⅓ that's easy to drop. Write the formula on the page before plugging in.