# 1 - Algebra

##### 2016-09-20 12:20:00 +0000, 2 years and 3 weeks ago

0 - Factorisation and Quadratic Formulae

Take for example the following:

In order to factorise it you must find which two numbers add up to -28, yet multiply to equal 13?
In this case it would be -2, -26, you then factorise them:

1 - Use and manipulation of surds

• Natural numbers: Positive integers (1, 2, 3 …)
• Integers: (-1, 0, 1 …)
• Rational numbers: “Any number that can be expressed in the form $\frac{p}{q}$ where p and q are integers and q is !=0 “
• Irrational numbers: Cannot be expressed as $\frac{p}{q}$ eg, $\sqrt{2}$

Rationalizing the denominator:

A surd is any number that contains a square root sign $\sqrt{x}$
We leave in a square root so that:

• We do not need to use a calculator when dealing with them.
• They are more accurate since we do not round them into a decimal.

For example $\sqrt{2}$ is an irrational number, it goes on for ever: $\sqrt{2} = 1.4142135623730950488...$

Simplification:
We simplfy surds by splitting them into 2 factors, where one of the factors is a square number.
(Square number: The product of a number multiplied by itself, e.g. 1, 4, 9, 16.) \

e.g.

$16$ is the biggest square factor, we could have used $4 \times 12$ but that would have not given use the most simplified answer.

Simplfying more complex surds:

In order to do this we must find a common factor between each surd, e.g.

Expanding dual bracket surds:
This in done the same way as a normal dual bracket is done, multiply the front, and then the back.
e.g.

Rationalising the denominator \

Take for example: $\frac{5}{\sqrt{3}}$, the denominator $\sqrt{3}$ is irrational, to simplify this we must remove the surd on the bottom.

e.g.

2 - Quadratic functions and their graphs

3 - The discriminant of a quadratic function

$ax^2+bx+c$ is descriminant

The expression $b^2-4ac$ is called the descriminant of the quadratic $ax^2+bx+c$

If $b^2-4ac > 0$ then there are 2 distinct roots.

If $b^2-4ac = 0$ then there is a repeating root.

If $% $ then there are no roots, and therefore there is no solution.

4 - Factorisation of quadratic polynomials

5 - Completing the square

Give $ax^2+bx+c$ in the form $(p+q)^2+z$

6 - Solution of quadratic equations

7 - Simultaneous Equations

8 - Solution of linear and quadratic inequalities

9 - Polynominals

10 - Simple algebraic division

11 - Remainder Theorum

12 - Factor Theorum

13 - Graphs of functions

14 - Geometric interpretation of algebraic solutions

15 - Graphical translations

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