Section 2 - Algebra Flashcards
Tell me some rules of negative numbers
+ + makes +
+ - makes -
- + makes -
- - makes +
Tell me about a minus number Getting squared
-3^2 is ambiguous
Should be written as (-3)^2 = 9
-(3^2) = -9
What is a term
A collection of numbers, letters and brackets, all multiplied/divided together
Eg x^2 term
4 is a number term
How can you simply or collecting like terms
Put bubbles around each term and its + or - sign
Then you can move the bubbles into the best order so that like terms are together
Combine like terms
Tell me the what to do when multiplying numbers with powers
Add the powers when the numbers are the same
Eg 3^6 X 3^4 = 3^10
What do you do when numbers with powers are being divided
Subtract the powers
What do you do when numbers with powers are being raised
Multiply them
(3^2)^4 = 2 X 4 = 8, 3^8
What happens when a power is 1
It’s just itself
eg 8^1 = 8
What is a number with the power of 0
It’s 1
5^0 = 1
What is 1 to any power
1
Eg 1^23 = 1
What do you do when fractions have a power
Apply the power to the top and bottom
(3/5)^3 = 3^3\5^3
What do you do when a number has a negative power
To make it into a number you turn it upside down to turn it into a positive power
Eg 7^-2 = 1/7^2
= 1/49
Tell me rules of fractional powers
The power 1/2 means square root
1/3 power means cube root
1/4 power means fourth root
When you get a negative power don’t forget to turn it upside down to make it positive then root it
What do you do with a 2 stage fractional power
Eg 64^5/6
Always split the fraction into a root and power and do root first then power
So (64)^1/6 X 5 = (64^1/6)^5
How do you multiply out single brackets
Remember to multiply each thing outside the bracket by each separate term inside
How do you multiply out double brackets
Have to multiply everything in the first bracket by everything in second bracket
Use eyebrows and Chins method
How do you multiply triple brackets out
Just multiply two together then multiply result by remaining bracket
Won’t be able to use eyebrows and Chins for last bracket so reduce it to single bracket multiplications
Tell me the method to factorise something into brackets for single brackets
Take out the biggest number that goes into all the terms
For each letter in term take out the highest power that will go into EVERY term
Open the bracket and fill in bits needed to reproduce each term
Check your answer by multiplying out the bracket and making sure it matches the original expression
When do you use DOTS
When you have one thing squared take away another thing square - the difference of 2 squares
Tell me how to use DOTS
Use this rule
A^2 - b^2 = (a + b)(a - b)
Tell me the 6 step method to solving equations
Get rid of any fractions
Multiply out any brackets
Collect all the X terms on one side and all the number terms on the other
Reduce it to the form ax = b by combining like terms
Divide by a to give X =
If you had x^2 = .. Instead sqaure root both sides to end up with X = - or + answer
What happens if you get x^2 = …
Square root it
But answer can be positive or negative!!!
How do you rearrange formulas with the solving equations method
Make something else the subject
Get rid of any square root signs by squaring both sides
Get rid of fractions
Multiply out any fractions
Collect all the subject terms on one side, all non subject terms on other
Reduce it to form ax=b
Divide both side by a to give X=
If your left with x^2 sqaure root it and remember it can be positive or negative
How do you factorise a quadratic
Put it into 2 brackets
Standard format is ax^2 + bx + c = 0
Tell me the factorising method when a = 1 (only x^2 )
Always rearrange into standard format
X^2 + bx + c = 0
Write down brackets (X )(X ) = 0
Find two numbers that multiply to the last number and add to the first
Multiply to give c and add/ subtract to give b
Fill in +/- signs to make sure they work
Solve equation by setting each bracket as equal to 0
How do you factorise quadratic when a is not 1
Eg 3x^2
Rearrange into standard form
Write down the initial brackets
Find pairs of numbers that multiply to give c - the last number
Try out number pairs so it works
Full in +/- signs
Check and solve equation by setting each bracket to 0
What’s the quadratic formula
X = -b +/- sqaure root b^2 - 4ac divided by 2a
Tell me the crucial details of the quadratic formula
Remember - Signs
Remember it’s 2a
- and divide all of top line by 2a
The +/- sign means you end up with 2 solutions
If you get a negative inside your sqaure root - go back and check your working - they won’t come up in GCSE
When do you use the quadratic formula
If you have a quadratic that won’t easily factorise
If the question mentions decimal places or significant figures
If the question asks for exacts answers or surds
How do you solve quadratics by completing the square
Rearrange quadratic into standard form ax^2 + bx + c
Write out initial bracket (X + b/2)^2 - just half b
Multiply out bracket and compare it to original to find what you need to add or subtract to complete the sqaure
Add or subtract the adjusting number to make it match the original
- value needed is always c - (b/2)
How do you complete the sqaure when a isn’t 1
Eg 3x^2
Follow same method but take out a factor of A from the x2 and X terms so you often divide b by an awkward fraction not 2
How does the completed sqaure help you sketch the graph
For a positive quadratic where the x^2 is positive, the adjusting number tells you the minimum brackets are equal to 0
The last bit in the bracket squares shows the minimum point in the graph and how far it is left or right is the number furthest right
- positive so won’t cross X axis
Find where curve crosses the y axis axis by substituting X= 0 into the equation
How do you simplify algebraic fractions
By cancelling terms on the top and bottom - deal with each letter individually
Might factorise too
How do you multiply or divide algebraic fractions
Multiply two fractions, just multiply tops and bottoms separately
To divide, turn the second fraction upside down and multiply
How do you add and subtract algebraic fractions
Work out a common denominator
Multiply top and bottom of each fraction by whatever gives you common denominator
Add or subtract numerators only
How do you find the nth term of a linear sequence
Works for linear sequences - ones with common differences
Where terms increase or decrease by same amounts
Also known as arithmetic sequence
Find the difference eg if it’s 3, 3n is the formula
Then work out what to add or subtract so for n=1, 3n is 3 so add 2 to get term 5 for sequence
Put both bits together
How do you find the nth term of a quadratic sequence
A quadratic sequence has an n^2 term
Difference between 2 terms changes
Find difference between each pair of terms
Work out the differences between the differences
Divide this value by 2
Subtract the n^2 term from each term to give you a linear sequence
Find the rule for the nth term of the linear sequence and add this on the n^2 term
N^2 + n + something
How do you decide if a term is in a sequence
Set the number equal to the nth term rule
And solve for n
If answer is not a whole number then it’s not in the sequence
Tell me about another type of sequence
Eg geometric progression - there is a common ratio where you multiply or divide by the same same number each time
How else can you write sequences
Written with u(little 1 in bottom right) for the first term and u(little 2 in bottom right) for second term and u(little n in bottom right) for nth term
U(little n+1) in bottom right is the next term
Tell me the inequality symbols
> greater than
< less than
> with a line means greater than or equal to
< with a line means less than or equal to
Tell me the exception to algebra with inequalities
Whenever you multiply or divide by a negative number you must FLIP THE INEQUALITY SIGN
Tell me about inequalities on number lines
An open circle like o is for < or > and a coloured in circle • is for < or > with lines
Tell me a general rule with quadratic inequalities
If x^2 > a^2 then X > a or X > -a
If x^2 < a^2 then -a < X < a
Quadratic inequalities means you will have 2 separate solutions
Tell me about quadratic inequality graphs
Factorise the graph, solve it the 2 answers are where it will cross the X axis and if the x^2 term is negative it will be n shaped
How do you show inequalities on a graph
Convert each inequality to an equation by putting an = in place of the inequality sings
Draw the graph for each equation
If inequality sign is < or > draw a dotted line but if it’s < or > with a line draw a solid line
Work out which side of line you want, put a point - usually origin to check
Shade the region this gives you
Whats an iterative method
Techniques where you keep repeating a calculation in order to get close to the solution you want
What’s important to remember about sign changes and iterative methods
If there’s a sign change (eg from positive to negative or vice versa) when you put two numbers into the equation, there’s a solution between those numbers
When do you use iteration
If you know an interval that contains a solution to an equation, use an iterative method to find the approximate value of solution
How do you use iterations
Try in order the values of X with 1dp that lies between the interval
Try entering the numbers into the equation with the 1dp numbers until there is a sign change - the solution will lie within those values
Try going to 2 dp
Tell me how to use an iteration machine
Start with x(bottom right n) and find value of X(little n+1) by using inverse of the equation, then enter that answer to equation to get X n+2 and so on
Continue until number doesn’t change
Tell me the 2 type of simultaneous equations
Easy ones where both are linear
tricky ones where ones quadratic
Tell me the 6 steps for simultaneous equations - easy ones
Rearrange both equations into the form
Ax + by = c
And label equations 1 and 2 match up the numbers so if you subtracted one, a coefficient - X of y would cancel out, multiply one or both of them by a suitable number
Add or subtract equations to eliminate terms with same coefficient
Solve to get remaining coefficient value
Substitute value into equation to get other coefficient value
Tell me the seven steps for tricky simultaneous equations
Rearrange quadratic equation (one with a squared something in it)
Substitute the quadratic expression into the other equation to get another equation
Rearrange to get a quadratic equation (two brackets) and solve - get 2 answers
Stick the first values back in one of the original equations - get 2 answers agian
Substitute both pairs of answers to check
Write out the pairs of answers
You will have 4
What are the solutions to simultaneous equations
Actsully coordinates of the points where the graphs of the equation cross
Quadratic is a u or n shape and other is a line
How do you shown things are odd or even
By multiples or rearranging
Even number can be 2n
Odd numbers 2n + 1
Consecutive numbers as n, n+1, n+2
The sum, difference and product of integers is always an integer
What’s the identity symbols
3 lines - an equals sign with an extra line
Means 2 things that are identically equal to each other
How can you disprove things
By finding a counter example
Find something that doesn’t work
What’s a function
Functions takes an input, processes it and outputs a value
You write f(X) = …
How are composite functions written
Combined functions
Fg(X) which means do G FIRST then F - do function closest to X
What are inverse functions
It’s the reverse
F^-1(X)
How do you find the inverse function
Write out the equation
X = f(y) ( y is the function)
Rearrange to make Y the subject
Replace y with f^-1(x)
