**Maths core 1**

Basic

Algebra

Manipulating

Algebraic

Expressions

Collecting like terms

Removing /expanding brackets

Factorising with one bracket

Multiplication

Algebraic fractions

Linear Equations and

Rearranging Formulae

Solve linear equations using

a variety of methods

Rearranging formula to

change the subject

e.g: make c the subject of:

Quadratic

Equations

**3 different types of quadratic equations:**

**1.**

**2.**

**3.**

**3 ways to**

solve

solve

**1. Factorising:**

**2. Completing the square:**

**e.g:**

**3. Quadratic Formula**

**e.g:**

Using the

discriminant

**(d)**

**and**

Simultaneous

Equations

Solve by elimination

Solve by substitution

1. Make coefficient of either y or x equal in both equations by multiplying the equation by the relevant number.

2. Subtract equation 2 from equation 1 to find the value of either y or x

3. Substitute the value into either equation 1 or equation 2 to find the value of the other variable.

1. If y isn’t already the subject, make y the subject of the equation

2. Substitute y

3. Solve the quadratic equation using preferred method.

Co-ordinate

Geometry

Equations of

Straight Lines

Passing through (0,c)

Passing through (x1,y1)

Straight line with equation:

ax+by=c

Passes through (c/a,0) and (0, c/b)

Gradients of

Straight Lines

Using point (a,b)

Gradient of Straight Line:

Need to know 2 points on the line

e.g for

Parallel lines have equal graideints

e.g:

y=2x+3

y=2x-1

Perpendicular lines:

Gradients multiply to give -1

m

x

m

= -1

Positive and negative gradients

Positive gradient

-e.g y=2x

Negative gradient

-e.g y=-2x

Distances and

Mid-points

-Pythagoras theorem

-Average

Circles and

other Curves

Equations of circles:

With centre (0,0) and radius r

With centre (a,b) and radius r

**Polynomials**

Curves and

Translations

**Uncertainties**

**Indices**

Operations with

Polynomials

Polynomial

functions

Factor and

Remainder

Theorems

Solving equations/

Graph sketching

Binomial Expansion

Quadratic

functions

Cubic

functions

Translations

**Working with**

Powers

and roots

Powers

and roots

**Laws of**

Indices

Indices

**Linear**

Inequalities

Inequalities

Points of intersection:

Finding the centre and radius from a sketch:

Adding Polynomials

Subtracting Polynomials

**Factor theorem**

**Remainder theorem**

Quadratic

Inequalities

**Rationalising the denominator**

**Simplifying square roots**

e.g

e.g

e.g

e.g

e.g

e.g

Solve by using simultaneous equation, substitute equation of other line or curve into equation of circle

e.g:

Centre: (

3

,

-4

) Radius:

6

Centre: midpoint of diameter

Radius: distance between centre and any other point on circle

Add together the terms

e.g

Subtract the terms

e.g

Multiplying Polynomials

Dividing Polynomials

Formulae

Example

hint

Always write it down in terms of a and b before substituting in values

Pascal's Triangle

(how you can find nCr)

Order/degree

Polynomials

As functions of x

Polynomial:

The power of x is an integer that is greater than or equal to 0

Distances:

A

B

Dx

Dy

(a)

(b)

Midpoint:

(x1,y1)

(x2,y2)

(a,b)=

A polynomial may be divided by linear functions. To divide a polynomial rearrange the equation so you are multiplying by the answer:

**Linear inequalities can be handeled just like linear equations, but when both sides are multiplied or divided by a negative number then the inequality must be reversed.**

**these can be solved by solving the corresponding quadratic equation, DRAWING the graph and then finding the solution set(s)**

3 different forms that the

equations can be in

1.

2.

3.

1. Finding the y intercept:

Finding points on a quadratic curve:

For:

e.g:

Graph:

Make x=0 and solve

So y intercept: (0,7)

2. Finding the root(s):

Make y=0

Expand brackets

Find a,b,c

Find d

Use the quadrtatic formula

to find the roots

So x intercepts: (-1,0) and (-7,0)

d<0 so 2 distinct roots

3. Finding the vertex (turning point):

So vertex: (-4,-9)

Line of symmetry: x=-4

Finding points on a quadratic curve:

For:

e.g:

Graph:

1. Finding the y intercept:

The value of c is the y intercept

+0

In this case y intercept: (0,0)

2. Finding the root(s):

Make y=0

Find a,b,c

Find d

Use the quadrtatic formula

to find the roots

So x intercepts: (0,0) and (-4,0)

3. Finding the vertex (turning point):

So vertex: (-2,-4)

Line of symmetry: x=-2

Finding points on a quadratic curve:

For:

e.g:

Graph:

1. Finding the y intercept:

Make x=0 and solve

So y intercept: (0,-4)

Look at the brackets,

the values that make them =0

are the x intercepts

So x intercepts: (-4,0) and (-1,0)

3. Finding the vertex (turning point):

So vertex: (-1.5,-6.25)

2. Finding the root(s):

Basic Parabolas:

When multiplying polynomials, draw out a grid and put in the polynomials, then multiply each term...

e.g:

x

...then add all the terms with the same order to getthe answer

So:

e.g:

So

By working backwards throught the table and sing multiplication you can work out the answer making sure that the answer to the multiplication is the same as the polynomial that is being divided

So:

Example of a cubic curve

Can be factorised using factor theorem to find one factor, then divide original equation by found factor to the the factor and a quadratic function, then factorise again to get the cubic function factorised. Then a graph can be sketched using that information. (see solvin equations/sketching graphs)

a<0

a>0