Official Practice Test · 2025 Edition

SAT Math
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20 Hard Questions · All Major Topics · Expert Solutions

20Questions
35minTimer
800Target Score


Core Concepts & Formulas
Memorize these before attempting the 20 hard questions

🔢 Algebra Essentials

Quadratic Formula
x = (−b ± √(b²−4ac)) / 2a
Discriminant
Δ = b²−4ac (Δ>0: 2 roots, Δ=0: 1 root, Δ<0: no real roots)
Vertex of Parabola
x = −b/2a, y = f(−b/2a)
Sum & Product of Roots
sum = −b/a, product = c/a
Difference of Squares
a²−b² = (a+b)(a−b)
Perfect Square
(a±b)² = a²±2ab+b²

📐 Geometry & Trig

Circle Area & Circumference
A = πr², C = 2πr
Arc Length
L = (θ/360)·2πr
Pythagorean Theorem
a²+b² = c²
SOH-CAH-TOA
sin=opp/hyp, cos=adj/hyp, tan=opp/adj
Special Right Triangles
30-60-90: 1:√3:2 | 45-45-90: 1:1:√2
sin²θ + cos²θ
= 1 (Pythagorean identity)

📊 Statistics & Data

Mean
x̄ = (Σxᵢ) / n
Percent Change
% = (new−old)/old × 100
Exponential Growth
A = P(1 + r)^t
Probability
P(A) = favorable / total

⭐ Must-Memorize Rules

  • When a quadratic has no real solutions, the discriminant b²−4ac < 0
  • Exponent rule: x^m · x^n = x^(m+n); (x^m)^n = x^(mn)
  • |ax + b| = c means ax+b = c OR ax+b = −c
  • For f(g(x)), substitute g(x) entirely into f(x)
  • Slope of perpendicular lines: m₁ × m₂ = −1
  • Complex number i² = −1; i³ = −i; i⁴ = 1
  • In similar triangles, corresponding sides are proportional
  • Remainder theorem: remainder of f(x)÷(x−k) equals f(k)

✏️ Worked Example

Q: If x² − 5x + 6 = 0, what are the values of x?
Factor: (x−2)(x−3) = 0 → x = 2 or x = 3

Q: A circle has area 36π. What is its circumference?
A = πr² = 36π → r = 6 → C = 2π(6) = 12π

Q: If sin θ = 3/5 and θ is acute, find cos θ.
sin²θ+cos²θ=1 → cos²θ=1−9/25=16/25 → cos θ=4/5
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