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Algebric formulas, Lecture notes of Applied Mathematics

Air University (AU)Applied Mathematics

basic algebric formulas

Typology: Lecture notes

2014/2015

Uploaded on 01/20/2015

azhar.iqbal
azhar.iqbal 🇬🇧

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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−2b+an−3b2+···+b
n−1)
10. an=a.a.a . . . n times
11. am.an=am+n
12. am
an=am−nif m>n
=1 ifm=n
=1
a
n−mif m<n;a∈R, a 6=0
13. (am)n=amn =(a
n
)
m
14. (ab)n=an.bn
15. a
bn=an
bn
16. a0=1wherea∈R, a 6=0
17. a−n=1
an,a
n=1
a
−n
18. ap/q =q
√ap
19. If am=anand a6=±1,a6=0thenm=n
20. If an=bnwhere n6=0,thena=±b
21. If √x, √yare quadratic surds and if a+√x=√y,thena= 0 and x=y
22. If √x, √yare quadratic surds and if a+√x=b+√ythen a=band x=y
23. If a, m, n are positive real numbers and a6=1,thenlog
amn =log
am+logan
24. If a, m, n are positive real numbers, a6=1,thenlog
am
n=log
am−logan
25. If aand mare positive real numbers, a6=1thenlog
am
n=nlogam
26. If a, b and kare positive real numbers, b6=1,k 6=1,thenlog
ba=logka
logkb
27. logba=1
logabwhere a, b are positive real numbers, a6=1,b 6=1
28. if a, m, n are positive real numbers, a6= 1 and if logam=log
a
n,then
m=n
Typese t by A
M
S-T
E
X
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MATHEMATICAL FORMULAE

Algebra

  1. (a + b)^2 = a^2 + 2ab + b^2 ; a^2 + b^2 = (a + b)^2 − 2 ab
  2. (a − b)^2 = a^2 − 2 ab + b^2 ; a^2 + b^2 = (a − b)^2 + 2ab
  3. (a + b + c)^2 = a^2 + b^2 + c^2 + 2(ab + bc + ca)
  4. (a + b)^3 = a^3 + b^3 + 3ab(a + b); a^3 + b^3 = (a + b)^3 − 3 ab(a + b)
  5. (a − b)^3 = a^3 − b^3 − 3 ab(a − b); a^3 − b^3 = (a − b)^3 + 3ab(a − b)
  6. a^2 − b^2 = (a + b)(a − b)
  7. a^3 − b^3 = (a − b)(a^2 + ab + b^2 )
  8. a^3 + b^3 = (a + b)(a^2 − ab + b^2 )
  9. an^ − bn^ = (a − b)(an−^1 + an−^2 b + an−^3 b^2 + · · · + bn−^1 )
  10. an^ = a.a.a... n times
  11. am.an^ = am+n
  12. a

m an^ =^ a

m−n (^) if m > n

= 1 if m = n = (^) an^1 −m if m < n; a ∈ R, a 6 = 0

  1. (am)n^ = amn^ = (an)m
  2. (ab)n^ = an.bn

( (^) a b

)n = a

n bn

  1. a^0 = 1 where a ∈ R, a 6 = 0
  2. a−n^ = (^) a^1 n , an^ = (^) a−^1 n
  3. ap/q^ = √qap
  4. If am^ = an^ and a 6 = ± 1 , a 6 = 0 then m = n
  5. If an^ = bn^ where n 6 = 0, then a = ±b
  6. If √x, √y are quadratic surds and if a + √x = √y, then a = 0 and x = y
  7. If √x, √y are quadratic surds and if a + √x = b + √y then a = b and x = y
  8. If a, m, n are positive real numbers and a 6 = 1, then loga mn = loga m+loga n
  9. If a, m, n are positive real numbers, a 6 = 1, then loga

( (^) m n

= loga m − loga n

  1. If a and m are positive real numbers, a 6 = 1 then loga mn^ = n loga m
  2. If a, b and k are positive real numbers, b 6 = 1, k 6 = 1, then logb a = log logk^ a k b
  3. logb a = (^) log^1 a b^

where a, b are positive real numbers, a 6 = 1, b 6 = 1

  1. if a, m, n are positive real numbers, a 6 = 1 and if loga m = loga n, then m = n

Typeset by AMS-TEX

2

  1. if a + ib = 0 where i = √−1, then a = b = 0
  2. if a + ib = x + iy, where i = √−1, then a = x and b = y
  3. The roots of the quadratic equation ax^2 +bx+c = 0; a 6 = 0 are −b^ ±

b^2 − 4 ac 2 a

The solution set of the equation is

−b +

2 a ,^

−b −

2 a

where ∆ = discriminant = b^2 − 4 ac

  1. The roots are real and distinct if ∆ > 0.
  2. The roots are real and coincident if ∆ = 0.
  3. The roots are non-real if ∆ < 0.
  4. If α and β are the roots of the equation ax^2 + bx + c = 0, a 6 = 0 then i) α + β = −ab = − (^) coeff. ofcoeff. of xx 2 ii) α · β = (^) ac = constant termcoeff. of x 2
  5. The quadratic equation whose roots are α and β is (x − α)(x − β) = 0 i.e. x^2 − (α + β)x + αβ = 0 i.e. x^2 − Sx + P = 0 where S =Sum of the roots and P =Product of the roots.
  6. For an arithmetic progression (A.P.) whose first term is (a) and the common difference is (d). i) nth^ term= tn = a + (n − 1)d ii) The sum of the first (n) terms = Sn = n 2 (a + l) = n 2 { 2 a + (n − 1)d} where l =last term= a + (n − 1)d.
  7. For a geometric progression (G.P.) whose first term is (a) and common ratio is (γ), i) nth^ term= tn = aγn−^1. ii) The sum of the first (n) terms:

Sn = a(1^ −^ γ

n) 1 − γ ifγ <^1 = a(γ

n (^) − 1) γ − 1 if^ γ >^1 = na if γ = 1

  1. For any sequence {tn}, Sn − Sn− 1 = tn where Sn =Sum of the first (n) terms.
  2. ∑n γ=

γ = 1 + 2 + 3 + · · · + n = n 2 (n + 1).

∑n γ=

γ^2 = 1^2 + 2^2 + 3^2 + · · · + n^2 = n 6 (n + 1)(2n + 1).