MATHEMATICAL FORMULAE
MATHEMATICAL FORMULAE
Algebra
1. (a + b)2 = a2 + 2ab + b2 ; a2 + b2 = (a + b)2 − 2ab
2. (a − b)2 = a2 − 2ab + b2 ; a2 + b2 = (a − b)2 + 2ab
3. (a + b + c)2 = a2 + b2 + c2 + 2(ab + bc + ca)
4. (a + b)3 = a3 + b3 + 3ab(a + b); a3 + b3 = (a + b)3 − 3ab(a + b)
5. (a − b)3 = a3 − b3 − 3ab(a − b); a3 − b3 = (a − b)3 + 3ab(a − b)
6. a2 − b2 = (a + b)(a − b)
7. a3 − b3 = (a − b)(a2 + ab + b2 )
8. a3 + b3 = (a + b)(a2 − ab + b2 )
9. an − bn = (a − b)(an−1 + an−2 b + an−3 b2 + · · · + bn−1 )
10. an = a.a.a . . . n times
11. am .an = am+n am 12. n = am−n if m > n a =1 if m = n
1
= n−m if m < n; a ∈ R, a = 0 a 13. (am )n = amn = (an )m
14. (ab)n = an .bn a n an 15.
= n b b
16. a0 = 1 where a ∈ R, a = 0
1
1
17. a−n = n , an = −n a a
√
18. ap/q = q ap
19. If am = an and a = ±1, a = 0 then m = n
20. If an = bn where n = 0, then a = ±b
√ √
√
√
21. If x, y are quadratic surds and if a + x = y, then a = 0 and x = y
√ √
√
√
22. If x, y are quadratic surds and if a + x = b + y then a = b and x = y
23. If a, m, n are positive real numbers and a = 1, then loga mn = loga m+loga n m 24. If a, m, n are positive real numbers, a = 1, then loga
= loga m − loga n n 25. If a and m are positive real numbers, a = 1 then loga mn = n loga m logk a
26. If a, b and k are positive real numbers, b = 1, k = 1, then logb a = logk b
1
27. logb a = where a, b are positive real numbers, a = 1, b = 1 loga b
28. if a, m, n are positive real numbers, a = 1 and if loga m = loga n, then m=n Typeset by AMS-TEX
2
29. if a + ib = 0
√
−1, then a = b = 0
√
where i = −1, then a = x and b = y
where i =
30. if a + ib = x + iy,
31. The roots of the quadratic equation ax2 +bx+c = 0; a = 0 are
The solution set of the equation is
−b ±
√ b2 − 4ac
2a
√
√
−b + ∆ −b − ∆
,
2a
2a
where ∆ = discriminant = b2 − 4ac
32. The roots are real and distinct if ∆ > 0.
33. The roots are real and coincident if ∆ = 0.
34. The roots are
Algebra
1. (a + b)2 = a2 + 2ab + b2 ; a2 + b2 = (a + b)2 − 2ab
2. (a − b)2 = a2 − 2ab + b2 ; a2 + b2 = (a − b)2 + 2ab
3. (a + b + c)2 = a2 + b2 + c2 + 2(ab + bc + ca)
4. (a + b)3 = a3 + b3 + 3ab(a + b); a3 + b3 = (a + b)3 − 3ab(a + b)
5. (a − b)3 = a3 − b3 − 3ab(a − b); a3 − b3 = (a − b)3 + 3ab(a − b)
6. a2 − b2 = (a + b)(a − b)
7. a3 − b3 = (a − b)(a2 + ab + b2 )
8. a3 + b3 = (a + b)(a2 − ab + b2 )
9. an − bn = (a − b)(an−1 + an−2 b + an−3 b2 + · · · + bn−1 )
10. an = a.a.a . . . n times
11. am .an = am+n am 12. n = am−n if m > n a =1 if m = n
1
= n−m if m < n; a ∈ R, a = 0 a 13. (am )n = amn = (an )m
14. (ab)n = an .bn a n an 15.
= n b b
16. a0 = 1 where a ∈ R, a = 0
1
1
17. a−n = n , an = −n a a
√
18. ap/q = q ap
19. If am = an and a = ±1, a = 0 then m = n
20. If an = bn where n = 0, then a = ±b
√ √
√
√
21. If x, y are quadratic surds and if a + x = y, then a = 0 and x = y
√ √
√
√
22. If x, y are quadratic surds and if a + x = b + y then a = b and x = y
23. If a, m, n are positive real numbers and a = 1, then loga mn = loga m+loga n m 24. If a, m, n are positive real numbers, a = 1, then loga
= loga m − loga n n 25. If a and m are positive real numbers, a = 1 then loga mn = n loga m logk a
26. If a, b and k are positive real numbers, b = 1, k = 1, then logb a = logk b
1
27. logb a = where a, b are positive real numbers, a = 1, b = 1 loga b
28. if a, m, n are positive real numbers, a = 1 and if loga m = loga n, then m=n Typeset by AMS-TEX
2
29. if a + ib = 0
√
−1, then a = b = 0
√
where i = −1, then a = x and b = y
where i =
30. if a + ib = x + iy,
31. The roots of the quadratic equation ax2 +bx+c = 0; a = 0 are
The solution set of the equation is
−b ±
√ b2 − 4ac
2a
√
√
−b + ∆ −b − ∆
,
2a
2a
where ∆ = discriminant = b2 − 4ac
32. The roots are real and distinct if ∆ > 0.
33. The roots are real and coincident if ∆ = 0.
34. The roots are