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Section 2 - Algebra Flashcards

1
Q

Tell me some rules of negative numbers

A

+ + makes +
+ - makes -
- + makes -
- - makes +

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2
Q

Tell me about a minus number Getting squared

A

-3^2 is ambiguous

Should be written as (-3)^2 = 9

-(3^2) = -9

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3
Q

What is a term

A

A collection of numbers, letters and brackets, all multiplied/divided together

Eg x^2 term
4 is a number term

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4
Q

How can you simply or collecting like terms

A

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

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5
Q

Tell me the what to do when multiplying numbers with powers

A

Add the powers when the numbers are the same

Eg 3^6 X 3^4 = 3^10

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6
Q

What do you do when numbers with powers are being divided

A

Subtract the powers

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7
Q

What do you do when numbers with powers are being raised

A

Multiply them

(3^2)^4 = 2 X 4 = 8, 3^8

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8
Q

What happens when a power is 1

A

It’s just itself

eg 8^1 = 8

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9
Q

What is a number with the power of 0

A

It’s 1

5^0 = 1

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10
Q

What is 1 to any power

A

1

Eg 1^23 = 1

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11
Q

What do you do when fractions have a power

A

Apply the power to the top and bottom

(3/5)^3 = 3^3\5^3

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12
Q

What do you do when a number has a negative power

A

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

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13
Q

Tell me rules of fractional powers

A

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

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14
Q

What do you do with a 2 stage fractional power

A

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

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15
Q

How do you multiply out single brackets

A

Remember to multiply each thing outside the bracket by each separate term inside

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16
Q

How do you multiply out double brackets

A

Have to multiply everything in the first bracket by everything in second bracket

Use eyebrows and Chins method

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17
Q

How do you multiply triple brackets out

A

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

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18
Q

Tell me the method to factorise something into brackets for single brackets

A

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

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19
Q

When do you use DOTS

A

When you have one thing squared take away another thing square - the difference of 2 squares

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20
Q

Tell me how to use DOTS

A

Use this rule

A^2 - b^2 = (a + b)(a - b)

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21
Q

Tell me the 6 step method to solving equations

A

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

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22
Q

What happens if you get x^2 = …

A

Square root it

But answer can be positive or negative!!!

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23
Q

How do you rearrange formulas with the solving equations method

Make something else the subject

A

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

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24
Q

How do you factorise a quadratic

A

Put it into 2 brackets

Standard format is ax^2 + bx + c = 0

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25
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
26
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
27
What's the quadratic formula
X = -b +/- sqaure root b^2 - 4ac divided by 2a
28
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
29
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
30
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)
31
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
32
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
33
How do you simplify algebraic fractions
By cancelling terms on the top and bottom - deal with each letter individually Might factorise too
34
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
35
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
36
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
37
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
38
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
39
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
40
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
41
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
42
Tell me the exception to algebra with inequalities
Whenever you multiply or divide by a negative number you must FLIP THE INEQUALITY SIGN
43
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
44
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
45
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
46
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
47
Whats an iterative method
Techniques where you keep repeating a calculation in order to get close to the solution you want
48
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
49
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
50
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
51
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
52
Tell me the 2 type of simultaneous equations
Easy ones where both are linear tricky ones where ones quadratic
53
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
54
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
55
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
56
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
57
What's the identity symbols
3 lines - an equals sign with an extra line Means 2 things that are identically equal to each other
58
How can you disprove things
By finding a counter example Find something that doesn't work
59
What's a function
Functions takes an input, processes it and outputs a value You write f(X) = ...
60
How are composite functions written
Combined functions Fg(X) which means do G FIRST then F - do function closest to X
61
What are inverse functions
It's the reverse F^-1(X)
62
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)

Maths (7 decks)

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