,Algebra
Quadratic Equations :
inequalities : eg-12-27 + 8 = 0
-
72 -
2x + 8 = 0
-
I(x2 + 2x
-
8) = 0
b = b2 49)
Formula : (((( 4)(x 2)
-
x =
+ 0
-
-
- =
2a -
I(x + 4)(x) -
2) - 0
CV'S :
-
4 ; 2
v I I 3
-
4 2
Exponential +
Logarithmic Equations :
*
keep the base on the left !
Exponent
j =
Logarithm
BP A BAP
eg) Determine inverses of
graph :
logs(x + 1)
1) <x 2 2)
y swap
= -
( y
-
Y
+ =
x =
7y -
2
TP ·
(x2 + 2) = 74 x =
10ys(y + 1)
log >
(x + 2)
y
=
8x =
y + 1
↳ shift gx
left
y
=
1
-
RUICS :
Exponents logs
amxan amth loya (AXB) logaatlogaB
=
*
* =
A aman =
am
-
n ~
loga (B/c) =
logaB-luga)
lamyn amn ~
(An) A
n loga
*
loga
= =
*
(b) m =
ampo *
luga (an)
=
n
*
(9/b) m =
am pm ↓
glogan = n
( (
*
90 =
/
*
loga
=
logb
A
a
-m = 1 am logb a
-ma im A
log D =
=
a
↓ Note so solution) and if it has
log equation terms
: -
if I
↳
L
unsolvable ! Calways chec though
If no base - make base 16
Simultaneous Equations
:
make = +
manipulate
, cubic Equations :
ax3 + ba + + d =
s
muT S Remember :
E
0
g()) where
-
F(x) intersect
& - =
1)
g(x)
solve f() above
by taking our +(1)
g(x]
0 +
>
commons
·
-
2) solve by inspection - matching F() gas 0 X t
·
+
-
Gue
.
(factor theorem
Nature of Roots :
W
discriminant X =
D2 -
492
(1x int 2) OO L01-Int) 3) 60 (2x int)
1) 0 =
0
-
-
↳
↳ real
equal , rational not real
imaginary -
perfect square
b
, :
,
real +rational
, unequal
not perfect square
:
-.
real ,
irrational +
unequal
Exponents and surds
1) Definitions / law : 2) Fractions with Exponents
x9xxb =
ya
+ b
(9yb =
xab Type 1 : when numerator and / or denominator
· find highest common factor
x9 x9
b
x" x9
-
=
=
ya
x
xb Y z(
+
5 3
2g
-
.
.
3 .
6
3
(xy)a x9y9 (18 1 34 31 5 split bases
= = =
-
-
. .
36
9 2
xy xY 34(3-5)
-
factorise
-
x =
1 HCF +
Find
=
=
-
29 x2 . 6
3
I
-
2
6
max(h
mn
, = = " x
1
=
simplify
-
>
-
" 3
xy =
"xxy S
3) Simple Equations Type 2 :
only I term in numerator and denominator
↳ '' by itself must be written prime
get ·
each base as
product of .
. zht ! (22)
n
+
1
eg eg .
31x # 24 (2 3)n .
-
1
.
x
=
34 =
zht .
z2n
-
2
>
- prime factors
1
x
84 2m 2n -1.zn
-
=
.
2
3r + y2n
-
=
22n
-
1
34-1 >
- the
. when bringing exponents to
ghH-nH 22n-2 + 1-2n
top change sign
=
.
- .
=
32 .
2-
=
9
2
&
Quadratic Equations :
inequalities : eg-12-27 + 8 = 0
-
72 -
2x + 8 = 0
-
I(x2 + 2x
-
8) = 0
b = b2 49)
Formula : (((( 4)(x 2)
-
x =
+ 0
-
-
- =
2a -
I(x + 4)(x) -
2) - 0
CV'S :
-
4 ; 2
v I I 3
-
4 2
Exponential +
Logarithmic Equations :
*
keep the base on the left !
Exponent
j =
Logarithm
BP A BAP
eg) Determine inverses of
graph :
logs(x + 1)
1) <x 2 2)
y swap
= -
( y
-
Y
+ =
x =
7y -
2
TP ·
(x2 + 2) = 74 x =
10ys(y + 1)
log >
(x + 2)
y
=
8x =
y + 1
↳ shift gx
left
y
=
1
-
RUICS :
Exponents logs
amxan amth loya (AXB) logaatlogaB
=
*
* =
A aman =
am
-
n ~
loga (B/c) =
logaB-luga)
lamyn amn ~
(An) A
n loga
*
loga
= =
*
(b) m =
ampo *
luga (an)
=
n
*
(9/b) m =
am pm ↓
glogan = n
( (
*
90 =
/
*
loga
=
logb
A
a
-m = 1 am logb a
-ma im A
log D =
=
a
↓ Note so solution) and if it has
log equation terms
: -
if I
↳
L
unsolvable ! Calways chec though
If no base - make base 16
Simultaneous Equations
:
make = +
manipulate
, cubic Equations :
ax3 + ba + + d =
s
muT S Remember :
E
0
g()) where
-
F(x) intersect
& - =
1)
g(x)
solve f() above
by taking our +(1)
g(x]
0 +
>
commons
·
-
2) solve by inspection - matching F() gas 0 X t
·
+
-
Gue
.
(factor theorem
Nature of Roots :
W
discriminant X =
D2 -
492
(1x int 2) OO L01-Int) 3) 60 (2x int)
1) 0 =
0
-
-
↳
↳ real
equal , rational not real
imaginary -
perfect square
b
, :
,
real +rational
, unequal
not perfect square
:
-.
real ,
irrational +
unequal
Exponents and surds
1) Definitions / law : 2) Fractions with Exponents
x9xxb =
ya
+ b
(9yb =
xab Type 1 : when numerator and / or denominator
· find highest common factor
x9 x9
b
x" x9
-
=
=
ya
x
xb Y z(
+
5 3
2g
-
.
.
3 .
6
3
(xy)a x9y9 (18 1 34 31 5 split bases
= = =
-
-
. .
36
9 2
xy xY 34(3-5)
-
factorise
-
x =
1 HCF +
Find
=
=
-
29 x2 . 6
3
I
-
2
6
max(h
mn
, = = " x
1
=
simplify
-
>
-
" 3
xy =
"xxy S
3) Simple Equations Type 2 :
only I term in numerator and denominator
↳ '' by itself must be written prime
get ·
each base as
product of .
. zht ! (22)
n
+
1
eg eg .
31x # 24 (2 3)n .
-
1
.
x
=
34 =
zht .
z2n
-
2
>
- prime factors
1
x
84 2m 2n -1.zn
-
=
.
2
3r + y2n
-
=
22n
-
1
34-1 >
- the
. when bringing exponents to
ghH-nH 22n-2 + 1-2n
top change sign
=
.
- .
=
32 .
2-
=
9
2
&
