Important Mathematical Formulas

(a + b)(a – b) = a² – b²

(a + b + c)² = a² + b² + c² + 2(ab + bc + ca)

(a ± b)² = a² + b²± 2ab

(a + b + c + d)² = a² + b² + c² + d² + 2(ab + ac + ad + bc + bd + cd)

(a ± b)³ = a³ ± b³ ± 3ab(a ± b)

(a ± b)(a² + b² m ab) = a³ ± b³

(a + b + c)(a² + b² + c² -ab – bc – ca) = a³ + b³ + c³ – 3abc = 

1/2 (a + b + c)[(a - b)² + (b - c)² + (c - a)²]

when a + b + c = 0, a³ + b³ + c³ = 3abc

(x + a)(x + b) (x + c) = x³ + (a + b + c) x² + (ab + bc + ac)x + abc

(x – a)(x – b) (x – c) = x³ – (a + b + c) x² + (ab + bc + ac)x – abc

a⁴ + a²b² + b⁴ = (a² + ab + b²)( a² – ab + b²)

a⁴ + b⁴ = (a² – √2ab + b²)( a² + √2ab + b²)

aⁿ + bⁿ = (a + b) (a ^n-1 – a ^n-2 b +  a ^n-3 b² – a ^n-4 b³ +…….. + b ^n-1)

(valid only if n is odd)

aⁿ – bⁿ = (a – b) (a ^n-1 + a ^n-2 b +  a ^n-3 b² + a ^n-4 b³ +……… + b ^n-1)

{where n ϵ N)

(a ± b)²ⁿ is always positive while -(a ± b)²ⁿ is always negative, for any real values of a and b

(a – b)²ⁿ = (b – a)²” and (a – b)²ⁿ⁺¹ = – (b – a)²ⁿ⁺¹

if α and β are the roots of equation ax² + bx + c = 0, roots of cx” + bx + a = 0 are 1/α and 1/β.
if α and β are the roots of equation ax² + bx + c = 0, roots of ax² – bx + c = 0 are -α and -β.

• n(n + l)(2n + 1) is always divisible by 6.

• 3²ⁿ leaves remainder = 1 when divided by 8


• n³ + (n + 1 )³ + (n + 2 )³ is always divisible by 9


• 10^2n + 1 + 1 is always divisible by 11


• n(n²- 1) is always divisible by 6


• n²+ n is always even


• 2³ⁿ-1 is always divisible by 7


• 15^2n-1 +l is always divisible by 16


• n³ + 2n is always divisible by 3


• 3⁴ⁿ – 4 ³ⁿ is always divisible by 17


• n! + 1 is not divisible by any number between 2 and n 

(where n! = n (n – l)(n – 2)(n – 3)…….3.2.1)

for eg 5! = 5.4.3.2.1 = 120 and similarly 10! = 10.9.8…….2.1= 3628800

Product of n consecutive numbers is always divisible by n!.

If n is a positive integer and p is a prime, then n^p – n is divisible by p.

|x| = x if x ≥ 0 and |x| = – x if x ≤ 0.

Minimum value of a².sec²Ɵ + b².cosec²Ɵ is (a + b)²; (0° < Ɵ < 90°) 

for eg. minimum value of 49 sec²Ɵ + 64.cosec²Ɵ is (7 + 8)² = 225.

among all shapes with the same perimeter a circle has the largest area.

if one diagonal of a quadrilateral bisects the other, then it also bisects the quadrilateral.

sum of all the angles of a convex quadrilateral = (n – 2)180°

number of diagonals in a convex quadrilateral = 0.5n(n – 3)

let P, Q are the midpoints of the nonparallel sides BC and AD of a trapezium ABCD.Then,
ΔAPD = ΔCQB.