Summary maths124

,CHAPTER 1 surds-irrational numbers expressed in terms of roots

ALGEBRAIC REVIEWS of rational numbers,eg. 2;1 we;us +




. Number systems unisation

N-counting N [1,2,3 3
(2Y) x 5
Natural numbers numbers 2 2
civs 2w
= = =
-
= -
.... +
4 -
3
I
Integers -
negative numbers, natural numbers and o
conjugate/recipicol:
change sigul
x E...
= -
3; -

2; -
1;0;1;2;33 Simplification
ene




Real numbers R 2 es0
+
2
=
+
Vz(25) 2 ve.
=

+
Yes 2
=

+
5e2

↳
Rational numbers - any number that can be written


in the form where by and a bak 1.3 logarithms
a [(a,bex
=



and b = 0 it is a positive number, then the logarithm of o


can be expressed in decimal form to the base b (b >0; b =1) denoted logy, is the

↳ such that bY
terminating 0.25, 0.625 numbery
=


x
eg.

↳
non-terminating, eg. 0,5 0,3333... 5 by xforx>0
y logp
= =
= =




exponent
↳non-terminating as fractions exponent 109 base Umber
=

base =number
m e




Letx=0.53
Logarithmic rules


(b 1)
logp
logbp.log
=


Because digits repeat, multiply both sides by 100 0
=

2



100x 63.53
=


logp log by then x
y
=
=




63.53 logb*
100x x 0.63
logp(y logyx logby x
- + =
- = =




99x 63 =



logp (5) logpx-logby =


lospa b
=




..
=

=
↳ Irrational numbers
M/Q -
a real number that


can'tbe expressed as a quotient of 2 integers

eg. 4,
5




1.2 Exponents andneeds
= -
x x .x .x .x ...)



The integer is the exponent (power or index) and x is the


base...." is ith power of o r to the power of a



Exponent laws

a0 1 =
a
=Va

a
c
u
=
~m
A =CaY* (a =
=
a
a.a* a
=
c r
+


=au- it a 0


a
=
authenm=n (a) a(ab) ab
= =

,CHAPTER 2 2.2 Solving equations

_
INEARB QUADRATIC EQUATIONS Factorisation
nee
cut
factors factors
2.1 Polynomials & rational expressions e.g.2 -3x 10 =
of a of c




Enterior
>I


1x2 3x 10 0
=
- -




x +2,


. x = -2 or S




Constant function
e m e re
f(x) b
=



use
the square

↳ horizontal line e.g.x2 - 6x - 5 0
=




"i
f(z) S =
x2 -
6x 3 =

move constant to RHS

f(s) 5
=

e
- 6x E) 5 (E)
+
=
+
add () to both sides

f( z) -
5
=


(x -
3)2 =

14 solve

x
-
3 =
=
V4 square root both sides


linear
mo
e nre
function f(x) mx+C
=


x =
=
x4 3
+

isolate s


↳
Straightline :x 44 3orx 44 3
= =
+ - +




Y
1
m-gradient/slope of the line
-y x
= +
0 > 0

unequal
1
2 -
the y-intercept
7
0 0
=
unequal
rational
- O




↳theslopeor astraightlinsPass through
case, it


2.3 Simultaneous equations

↳ if L,822 lines with
are 2 non-vertical respectto Isolate a variable in the simpler equation

slopes mime, then substitute the equivalent expression for that variable

Li is
perpendicular to 12 when m,xmz=-1 into the other expression

hi parallel
is to 12 when m2
mi = Solve the
newly formed equation (note there is
only
one variable now




Que
function f(x) acch + bx
=

+ Substitute your solution(s) back into the other

↳ parabola order to find the value (5)
equation in corresponding of


the other value
P(x)
q(x) =
0
umexpressions a()

, CHAPTER 3 Graph the following as
single intervals
and

INEQUALITIES & ABSOLUTEVALUES
/


eg.(3;10]1(8;d) (4;83U[S,0)
⑳ - ⑧
3.1 Intervals 3

0 ⑧ 8
0

An interval is set of real numbers between
a
given b j
>x
lo
>
↓ ' j
numbers

↳ Interval notation :x[[8;10] ..x(4;)



3 1958
Round brackets and open circles indicate the


number is excluded while square brackets and 3.2 Absolute values

closed circles indicate the number is included the a real number
absolute value of is the distance of a



↳ Set
builder notation number on the real line (number line) from zero. So, the absolve


used to represent all sers because the set is specified value is always positive

>less than 4 less than or
equal to
Distance is positive

E
I I -
than than equal ka1 if 0
greater greater to x x
=
or


-
I
ic
↓

S12 = 2;EIR if
-
-
x co



real numbers greater than -3 and less and
represents

equal to 2 properties
m



Properties
b e
r ee n
of
inequalities ↓a) (a) =
the distance from zero to -
as a is equal


ifa s and by then as c lab) 1allbl
=




I
ifabb d
12
and cd then a c-b
+
+ =
it by o

it as and 330 then ac<bC

but ifabb and CC0 then ac <bc |a b) (a) 1b)
+
+
This is know as the triangle inequality

use
equalities
when
multiply or
dividing both sides of an Solving inequalities
em
a reere
to
less than or equal

number the direction
↳ When Ixtal Ib b3··
inequality by
=
of the the -b =x+9
a
negative

inequality sign changes eg.
12x 1) +
[
3

=
3 2x 1 =
Adding and subtracting doesn't effect the inequality 3
-
+




=
sign. -
4 2x 2


Never divide or
multiply both sides of an
inequality -
2[x[1.xd( 2;1] -




unknown
by an


↳
when lotal- 3




D
...
No solution

sand von