MU123 Unit 9 Notes
1 Number patterns and algebra
1.1 Arithmetic sequences
Sum of a sequence of natural numbers can be rewritten as a sum of pairs of numbers:
In case of a sum of successive odd numbers the sum is always the square of how many odd numbers I add, e.g.
Formula for
Any list of numbers is called a sequence
is finite sequence is infinite sequence
The numbers in a sequence are called the terms of the sequence.
Arithmetic sequence is a sequence where the difference between consequtive terms is constant, e.g.:
The
The number of terms
The sum of the finite arithmetic sequence with first term
2 Multiplying out pairs of brackets
2.1 Pairs of brackets
Strategy to multiply out two brackets
Multiply each term inside the first bracket by each term inside the second bracket, and add the resulting terms.
2.2 Squaring brackets
Hence the general formula:
2.3 Differences of two squares
3 Quadratic expressions and equations
3.1 Quadratic expressions
An expression of the form
3.2 Quadratic equations
An equation that can be expressed in the form:
3.3 Solving simple quadratic equations
An equation of the form
Equations as
3.4 Factorising quadratics of the form
Fill in the gaps in the brackets on the right-hand side of the equation
3.5 Solving quadratic equations by factorisation
If the product of two or more numbers is 0, then at least one of the numbers must be 0.
Hence in an equation as
Strategy to solve by factorisation
- Find a factorisation:
- Then
, so or and hence the solutions are and .
When the two solutions are same it's said that equation has a repeated solution.
3.6 Factorising quadratics of the form
E.g. following quadratic expression;
Find two numbers whose product is
and whose sum is .
The numbers are
Rewrite the quadratic expression, splitting the term in
using the above factor pair.
Group the four terms in pairs and take out common factors to give the required factorisation.
Simplify a quadratic equation
- If the coefficient of
is negative, then multiply the equation through by to make this coefficient positive. - If the coefficients have a common factor, then divide the equation through by this factor.
- If any of the coefficients are fractions, then multiply the equation through by a suitable number to clear them.