Algebra

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. Ifan = 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. Ifa, 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
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29. if a + ib = 0

where i =

30. if a + ib = x + iy,

√ −1, then a = b = 0 √ where i = −1, thena = x and b = y √ √ −b + ∆ −b − ∆ , 2a 2a

31. The roots of the quadratic equation ax2 +bx+c = 0; a = 0 are

−b ±

√ b2 − 4ac 2a

The solution set of the equation is 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 non-real if ∆ < 0. 35. If α and β are the roots of the equation ax2 + bx +...