High School Mathematics - Quick Notes
Most important formulas and rules for fast revision
1. Algebra & Indices
Topic Formula / Rule
Law of indices am x an = am+n; am / an = am-n; (am)n = amn
Special indices a0 = 1; a-n = 1 / an; a1/n = n-th root of a
Square of sum (a + b)2 = a2 + 2ab + b2
Square of difference (a - b)2 = a2 - 2ab + b2
Difference of squares a2 - b2 = (a + b)(a - b)
Cube of sum (a + b)3 = a3 + 3a2b + 3ab2 + b3
Sum / diff of cubes a3 +- b3 = (a +- b)(a2 -+ ab + b2)
2. Quadratic Equations
• Standard form: ax2 + bx + c = 0. Solve by factorisation, completing the square or the quadratic formula.
• Discriminant D = b2 - 4ac. If D > 0: two real roots; D = 0: equal roots; D < 0: no real roots.
Topic Formula / Rule
Quadratic formula x = (-b +- sqrt(b2 - 4ac)) / 2a
Sum of roots α + β = -b / a
Product of roots αβ = c / a
Vertex of parabola x = -b / 2a
3. Sets, Logarithms & Sequences
Topic Formula / Rule
Log laws log(AB) = log A + log B; log(A/B) = log A - log B; log An = n log A
Definition ax = N means loga N = x
Arithmetic progression n-th term: a + (n-1)d; sum: S = (n/2)[2a + (n-1)d]
Geometric progression n-th term: arn-1; sum: S = a(rn - 1)/(r - 1)
Sets n(A or B) = n(A) + n(B) - n(A and B)
4. Geometry & Mensuration
• Pythagoras: in a right triangle, c2 = a2 + b2. Angles in a triangle add to 180 deg; in a quadrilateral 360
deg. Angle in a semicircle = 90 deg.
• Similar triangles have equal angles and proportional sides. Congruence tests: SSS, SAS, ASA, RHS.
High School Mathematics - Quick Notes | Page 1
Most important formulas and rules for fast revision
1. Algebra & Indices
Topic Formula / Rule
Law of indices am x an = am+n; am / an = am-n; (am)n = amn
Special indices a0 = 1; a-n = 1 / an; a1/n = n-th root of a
Square of sum (a + b)2 = a2 + 2ab + b2
Square of difference (a - b)2 = a2 - 2ab + b2
Difference of squares a2 - b2 = (a + b)(a - b)
Cube of sum (a + b)3 = a3 + 3a2b + 3ab2 + b3
Sum / diff of cubes a3 +- b3 = (a +- b)(a2 -+ ab + b2)
2. Quadratic Equations
• Standard form: ax2 + bx + c = 0. Solve by factorisation, completing the square or the quadratic formula.
• Discriminant D = b2 - 4ac. If D > 0: two real roots; D = 0: equal roots; D < 0: no real roots.
Topic Formula / Rule
Quadratic formula x = (-b +- sqrt(b2 - 4ac)) / 2a
Sum of roots α + β = -b / a
Product of roots αβ = c / a
Vertex of parabola x = -b / 2a
3. Sets, Logarithms & Sequences
Topic Formula / Rule
Log laws log(AB) = log A + log B; log(A/B) = log A - log B; log An = n log A
Definition ax = N means loga N = x
Arithmetic progression n-th term: a + (n-1)d; sum: S = (n/2)[2a + (n-1)d]
Geometric progression n-th term: arn-1; sum: S = a(rn - 1)/(r - 1)
Sets n(A or B) = n(A) + n(B) - n(A and B)
4. Geometry & Mensuration
• Pythagoras: in a right triangle, c2 = a2 + b2. Angles in a triangle add to 180 deg; in a quadrilateral 360
deg. Angle in a semicircle = 90 deg.
• Similar triangles have equal angles and proportional sides. Congruence tests: SSS, SAS, ASA, RHS.
High School Mathematics - Quick Notes | Page 1
