Algebra formulae

1. MATHEMATICAL FORMULAE
Algebra
1. (a+b)
2
=a
2
+2ab+b
2
;a
2
+b
2
=(a+b)
2
2ab
2. (ab)
2
=a
2
2ab+b
2
;a
2
+b
2
=(ab)
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
3ab(a+b)
5. (ab)
3
=a
3
b
3
3ab(ab);a
3
b
3
=(ab)
3
+3ab(ab)
6.a
2
b
2
=(a+b)(ab)
7.a
3
b
3
=(ab)(a
2
+ab+b
2
)
8.a
3
+b
3
=(a+b)(a
2
ab+b
2
)
9.a
n
b
n
=(ab)(a
n1
+a
n2
b+a
n3
b
2
++b
n1
)
10.a
n
=a:a:a : : : ntimes
11.a
m
:a
n
=a
m+n
12.
a
m
a
n
=a
mn
ifm>n
=1 if m=n
=
1
a
nm
ifm<n;a2R; a6 =0
13. (a
m
)
n
=a
mn
=(a
n
)
m
14. (ab)
n
=a
n
:b
n
15.

a b

n
=
a
n
b
n
16.a
0
=1wherea2R; a6 =0
17.a
n
=
1 a
n
;a
n
=
1
a
n
18.a
p=q
=
q
p
a
p
19. Ifa
m
=a
n
anda6 =1;a6 =0thenm=n
20. Ifa
n
=b
n
wheren6 =0,thena=b
21. If
p
x;
p
yare quadratic surds and ifa+
p
x=
p
y,thena= 0 andx=y
22. If
p
x;
p
yare quadratic surds and ifa+
p
x=b+
p
ythena=bandx=y
23. Ifa; m; nare positive real numbers anda6 =1,thenlog
amn=log
am+log
an
24. Ifa; m; nare positive real numbers,a6 =1,thenlog
a

m
n

=log
amlog
an
25. Ifaandmare positive real numbers,a6 =1thenlog
am
n
=nlog
am
26. Ifa; bandkare positive real numbers,b6 =1;k6 =1,thenlog
ba=
log
ka
log
kb
27. log
ba=
1
log
ab
wherea; bare positive real numbers,a6 =1;b6 =1
28. ifa; m; nare positive real numbers,a6 = 1 and if log
am=log
an,then
m=n
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2. 2
29. ifa+ib=0 wherei=
p
1, thena=b=0
30. ifa+ib=x+iy,wherei=
p
1, thena=xandb=y
31. The roots of the quadratic equationax
2
+bx+c=0;a6 = 0 are
b
p
b
2
4ac
2a
The solution set of the equation is
(
b+
p

2a
;
b
p

2a
)
where = discriminant =b
2
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. Ifandare the roots of the equationax
2
+bx+c=0;a6 =0then
i)+=
b
a
=
coe. ofx
coe. ofx
2
ii)=
c
a
=
constant term
coe. ofx
2
36. The quadratic equation whose roots areandis (x)(x)=0
i.e.x
2
(+)x+=0
i.e.x
2
Sx+P=0whereS=Sum of the roots andP=Product of the
roots.
37. For an arithmetic progression (A.P.) whose rst term is (a) and the common
dierence is (d).
i)n
th
term=t n=a+(n1)d
ii) The sum of the rst (n)terms=S
n=
n
2
(a+l)=
n
2
f2a+(n1)dg
wherel=last term=a+(n1)d.
38. For a geometric progression (G.P.) whose rst term is (a) and common ratio
is (),
i)n
th
term=t n=a
n1
.
ii) The sum of the rst (n)terms:
S
n=
a(1
n
)
1
if<1
=
a(
n
1)
1
if>1
=na if=1
:
39. For any sequenceft
ng;SnSn1=tnwhereS n=Sum of the rst (n)
terms.
40.
nP
=1
=1+2+3+ +n=
n
2
(n+1).
41.
nP
=1

2
=1
2
+2
2
+3
2
++n
2
=
n
6
(n+ 1)(2n+1).

3. 3
42.
nP
=1

3
=1
3
+2
3
+3
3
+4
3
++n
3
=
n
2
4
(n+1)
2
.
43.n!=(1):(2):(3):::::(n1):n.
44.n!=n(n1)! =n(n1)(n2)! =::::.
45. 0! = 1.
46. (a+b)
n
=a
n
+na
n1
b+
n(n1)
2!
a
n2
b
2
+
n(n1)(n2)
3!
a
n3
b
3
++
b
n
;n>1.