ALEKS Math Placement Cheat Sheet.

Percents & Proportions

• Percent change: (new - old) / old × 100. Always divide by the original value.
• Percent of a number: rate as a decimal × base. 35% of 240 = 0.35 × 240 = 84.
• Original price after discount: original = final ÷ (1 - rate). Paid 68 after 20% off: 68 ÷ 0.80 = 85.
• Proportion: a/b = c/d means ad = bc. Pair each numerator with the opposite denominator.
• Successive percents multiply: 12% off then 8% tax = × 0.88 × 1.08, never a flat -4%.

Geometry

• Rectangle: A = lw, P = 2(l + w). Triangle: A = (1/2)bh, angles sum to 180.
• Circle: A = πr^2, C = πd = 2πr. Square the radius, not the diameter.
• Box: V = lwh. Cylinder: V = πr^2h.
• Pythagorean theorem: a^2 + b^2 = c^2. Triples: 3-4-5, 5-12-13, 8-15-17.
• Complementary angles sum to 90; supplementary to 180.

Linear Equations & Lines

• Slope: m = (y2 - y1)/(x2 - x1). Slope-intercept: y = mx + b. Point-slope: y - y1 = m(x - x1).
• Parallel: equal slopes. Perpendicular: slopes multiply to -1 (negative reciprocals).
• Intercepts: x-intercept sets y = 0; y-intercept sets x = 0.
• Inequality flip: reverse the sign only when multiplying or dividing by a negative.
• Absolute value: |X| = k (k > 0) splits into X = k or X = -k.

Exponent Laws

• x^a · x^b = x^(a+b); x^a ÷ x^b = x^(a-b); (x^a)^b = x^(ab).
• x^0 = 1 (x ≠ 0); x^(-n) = 1/x^n; x^(m/n) = (nth root of x)^m. 16^(3/4) = 8.
• (2a^3)^4 = 16a^12: raise every factor, multiply the exponents.

Factoring Patterns

• Difference of squares: a^2 - b^2 = (a - b)(a + b).
• Perfect square: (a ± b)^2 = a^2 ± 2ab + b^2. Never drop the middle term.
• Trinomial x^2 + bx + c: find two numbers with product c and sum b.
• Always pull the GCF first: 2x^2 + 10x + 12 = 2(x + 2)(x + 3).

Quadratics, Exponentials & Logs

• Quadratic formula: x = (-b ± √(b^2 - 4ac)) / (2a).
• Discriminant b^2 - 4ac: positive → 2 real roots; zero → 1; negative → none.
• Vertex: x = -b/(2a); substitute back for y.
• Log meaning: log_b(x) = y means b^y = x.
• Log rules: log(ab) = log a + log b; log(a/b) = log a - log b; log(a^n) = n log a.
• Same-base equations: b^m = b^n forces m = n. Check log arguments stay positive.

Trigonometry

• SOH-CAH-TOA: sin = opp/hyp, cos = adj/hyp, tan = opp/adj.
• Special values: sin 30 = 1/2, cos 30 = √3/2, sin 45 = cos 45 = √2/2, tan 45 = 1, sin 60 = √3/2, cos 60 = 1/2.
• Radians: degrees × π/180; 180° = π; 150° = 5π/6.
• Identity: sin^2 θ + cos^2 θ = 1, so 1 - cos^2 θ = sin^2 θ.
• Quadrant signs: all positive in QI; sin in QII; tan in QIII; cos in QIV.