# Important Mathematical Formulas

## Maths Formulas

**(a + b)(a – b) = a**^{2}– b^{2}

**(a + b + c)**^{2}= a^{2}+ b^{2}+ c^{2}+ 2(ab + bc + ca)

**(a ± b)**^{2}= a^{2}+ b^{2}± 2ab

**(a + b + c + d)**^{2}= a^{2}+ b^{2}+ c^{2}+ d^{2}+ 2(ab + ac + ad + bc + bd + cd)

**(a ± b)**^{3}= a^{3}± b^{3}± 3ab(a ± b)

**(a ± b)(a**^{2}+ b^{2}m ab) = a^{3}± b^{3}

**(a + b + c)(a**^{2}+ b^{2}+ c^{2}-ab – bc – ca) = a^{3}+ b^{3}+ c^{3}– 3abc =

**1/2 (a + b + c)[(a – b)**^{2}+ (b – c)^{2}+ (c – a)^{2}]

**when a + b + c = 0, a**^{3}+ b^{3}+ c^{3}= 3abc

**(x + a)(x + b) (x + c) = x**^{3}+ (a + b + c) x^{2}+ (ab + bc + ac)x + abc

**(x – a)(x – b) (x – c) = x**^{3}– (a + b + c) x^{2}+ (ab + bc + ac)x – abc

**a**^{4}+ a^{2}b^{2}+ b^{4}= (a^{2}+ ab + b^{2})( a^{2}– ab + b^{2})

**a**^{4}+ b^{4}= (a^{2}– √2ab + b^{2})( a^{2}+ √2ab + b^{2})

**a**^{n}+ b^{n}= (a + b) (a^{n-1}– a^{n-2}b + a^{n-3}b^{2}– a^{n-4}b^{3}+…….. + b^{n-1})

** (valid only if n is odd)**

**a**^{n}– b^{n}= (a – b) (a^{n-1}+ a^{n-2}b + a^{n-3}b^{2}+ a^{n-4}b^{3}+……… + 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 ax**^{2}+ bx + c = 0, roots of cx” + bx + a = 0 are 1/α and 1/β.

if α and β are the roots of equation ax^{2}+ bx + c = 0, roots of ax^{2}– bx + c = 0 are -α and -β.

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

**3**^{2n}leaves remainder = 1 when divided by 8

**n**^{3}+ (n + 1 )^{3}+ (n + 2 )^{3}is always divisible by 9

**10**^{2n}^{+}^{1}+ 1 is always divisible by 11

**n(n**^{2}- 1) is always divisible by 6

**n**^{2}+ n is always even

**2**^{3n}-1 is always divisible by 7

**15**^{2n-1 }+l is always divisible by 16

**n**^{3}+ 2n is always divisible by 3

**3**^{4n}– 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 a**^{2}.sec^{2}Ɵ + b^{2}.cosec^{2}Ɵ is (a + b)^{2}; (0° < Ɵ < 90°)

**for eg. minimum value of 49 sec**^{2}Ɵ + 64.cosec^{2}Ɵ 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.