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ACT Mathematics Advanced Problem Solving
25 flashcards covering advanced algebra, coordinate geometry, trigonometry identities, and statistics for the ACT Math test.
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Quadratic Formula
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1
Quadratic Formula
x = (-b +- sqrt(b^2 - 4ac)) / (2a); finds roots of any quadratic equation ax^2 + bx + c = 0.
2
Discriminant Interpretation
D = b^2 - 4ac; D > 0 gives two real roots, D = 0 gives one real repeated root, D < 0 gives two complex conjugate roots.
3
Vieta's Formulas for Quadratics
For ax^2 + bx + c = 0, sum of roots r1 + r2 = -b/a, and product of roots r1 * r2 = c/a.
4
Slope of a Line Between Two Points
m = (y2 - y1) / (x2 - x1); vertical rise divided by horizontal run across points (x1, y1) and (x2, y2).
5
Slopes of Parallel and Perpendicular Lines
Parallel lines have identical slopes (m1 = m2); perpendicular lines have negative reciprocal slopes (m1 * m2 = -1).
6
Midpoint Formula in Cartesian Coordinates
M = ((x1 + x2) / 2, (y1 + y2) / 2); calculates the exact middle point of a line segment.
7
Distance Formula in Coordinate Plane
d = sqrt((x2 - x1)^2 + (y2 - y1)^2); derived directly from the Pythagorean theorem.
8
Standard Equation of a Circle
(x - h)^2 + (y - k)^2 = r^2; defines a circle with center (h, k) and radius r in Cartesian coordinates.
9
Primary Trigonometric Ratios (SOH CAH TOA)
sin(theta) = opp/hyp, cos(theta) = adj/hyp, tan(theta) = opp/adj in any right-angled triangle.
10
Fundamental Pythagorean Trigonometric Identity
sin^2(theta) + cos^2(theta) = 1; valid for all real angles theta.
11
Tangent Quotient Identity
tan(theta) = sin(theta) / cos(theta); undefined wherever cos(theta) = 0.
12
Law of Sines for Oblique Triangles
sin(A)/a = sin(B)/b = sin(C)/c; relates angles to opposite sides in any non-right triangle.
13
Law of Cosines Formula
c^2 = a^2 + b^2 - 2ab*cos(C); generalizes the Pythagorean theorem to any triangle.
14
Arithmetic Sequence n-th Term Formula
a_n = a_1 + (n - 1)*d; where a_1 is the initial term and d is the common difference.
15
Geometric Sequence n-th Term Formula
a_n = a_1 * r^(n - 1); where a_1 is the first term and r is the common ratio.
16
Sum of a Finite Arithmetic Series
S_n = (n / 2) * (a_1 + a_n); average of first and last terms multiplied by number of terms n.
17
Logarithm Product and Quotient Rules
log_b(xy) = log_b(x) + log_b(y), and log_b(x/y) = log_b(x) - log_b(y).
18
Logarithm Power Rule
log_b(x^k) = k * log_b(x); exponents inside logarithms can be brought out as multiplying coefficients.
19
Powers of Imaginary Unit i Cycle
i = sqrt(-1), i^2 = -1, i^3 = -i, i^4 = 1; pattern repeats every four powers (evaluate modulo 4).
20
Matrix Multiplication Dimension Requirement
Matrix A (m x k) and Matrix B (k x n) can multiply because inner dimensions match, resulting in size m x n.
21
Probability of Independent Compound Events
P(A and B) = P(A) * P(B); joint probability equals the product of individual probabilities.
22
Probability of Mutually Exclusive Events
P(A or B) = P(A) + P(B); addition rule when events share no common outcomes.
23
Combinations Formula (Order Does Not Matter)
nCr = n! / (r! * (n - r)!); selections where the arrangement order of chosen items does not matter.
24
Permutations Formula (Order Matters)
nPr = n! / (n - r)!; arrangements where the order of selected items creates distinct outcomes.
25
Measures of Central Tendency: Mean, Median, Mode
Mean = sum/n; Median = middle value of sorted list; Mode = most frequently occurring observation.