Important Mathematical Formulas

1. (a + b)(a – b) = a2 – b2

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

3. (a ± b)2 = a2 + b2± 2ab

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

5. (a ± b)3 = a3 ± b3 ± 3ab(a ± b)

6. (a ± b)(a2 + b2 m ab) = a3 ± b3

7. (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]

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

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

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

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

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

13. 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)

14. 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)

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

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

17. 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.

18.
* 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

* 102n + 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

* 152n-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

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

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

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

22. 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.

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

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

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

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

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

If you find any error in above equations then please inform me.