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Maths > Section 2 - Algebra > Flashcards

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
Q

Tell me the factorising method when a = 1 (only x^2 )

A

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
Q

How do you factorise quadratic when a is not 1

Eg 3x^2

A

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
Q

What’s the quadratic formula

A

X = -b +/- sqaure root b^2 - 4ac divided by 2a

28
Q

Tell me the crucial details of the quadratic formula

A

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
Q

When do you use the quadratic formula

A

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
Q

How do you solve quadratics by completing the square

A

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
Q

How do you complete the sqaure when a isn’t 1

Eg 3x^2

A

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
Q

How does the completed sqaure help you sketch the graph

A

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
Q

How do you simplify algebraic fractions

A

By cancelling terms on the top and bottom - deal with each letter individually

Might factorise too

34
Q

How do you multiply or divide algebraic fractions

A

Multiply two fractions, just multiply tops and bottoms separately

To divide, turn the second fraction upside down and multiply

35
Q

How do you add and subtract algebraic fractions

A

Work out a common denominator

Multiply top and bottom of each fraction by whatever gives you common denominator

Add or subtract numerators only

36
Q

How do you find the nth term of a linear sequence

A

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
Q

How do you find the nth term of a quadratic sequence

A

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
Q

How do you decide if a term is in a sequence

A

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
Q

Tell me about another type of sequence

A

Eg geometric progression - there is a common ratio where you multiply or divide by the same same number each time

40
Q

How else can you write sequences

A

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
Q

Tell me the inequality symbols

A

> greater than
< less than

> with a line means greater than or equal to
< with a line means less than or equal to

42
Q

Tell me the exception to algebra with inequalities

A

Whenever you multiply or divide by a negative number you must FLIP THE INEQUALITY SIGN

43
Q

Tell me about inequalities on number lines

A

An open circle like o is for < or > and a coloured in circle • is for < or > with lines

44
Q

Tell me a general rule with quadratic inequalities

A

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
Q

Tell me about quadratic inequality graphs

A

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
Q

How do you show inequalities on a graph

A

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
Q

Whats an iterative method

A

Techniques where you keep repeating a calculation in order to get close to the solution you want

48
Q

What’s important to remember about sign changes and iterative methods

A

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
Q

When do you use iteration

A

If you know an interval that contains a solution to an equation, use an iterative method to find the approximate value of solution

50
Q

How do you use iterations

A

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
Q

Tell me how to use an iteration machine

A

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
Q

Tell me the 2 type of simultaneous equations

A

Easy ones where both are linear

tricky ones where ones quadratic

53
Q

Tell me the 6 steps for simultaneous equations - easy ones

A

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
Q

Tell me the seven steps for tricky simultaneous equations

A

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
Q

What are the solutions to simultaneous equations

A

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
Q

How do you shown things are odd or even

A

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
Q

What’s the identity symbols

A

3 lines - an equals sign with an extra line

Means 2 things that are identically equal to each other

58
Q

How can you disprove things

A

By finding a counter example

Find something that doesn’t work

59
Q

What’s a function

A

Functions takes an input, processes it and outputs a value

You write f(X) = …

60
Q

How are composite functions written

A

Combined functions

Fg(X) which means do G FIRST then F - do function closest to X

61
Q

What are inverse functions

A

It’s the reverse

F^-1(X)

62
Q

How do you find the inverse function

A

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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