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GRE MATHS > Algebra > Flashcards

Algebra Flashcards

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1
Q

ca+cb =

A

ca+cb = c(a+b)

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2
Q

ca-cb =

A

ca-cb = c (a - b)

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3
Q

(a + b)² =

A

(a + b)² = a² + 2ab + b²

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4
Q

(a - b)² =

A

(a - b)² = a² - 2ab + b2

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5
Q

a² - b² =

A

a² - b² = (a+b)(a-b)

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6
Q

(a + b)³ =

A

(a+b)³ = a³ + 3a²b + 3ab² - b³

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7
Q

(a - b)³ =

A

(a - b)³ = a³ - 3a²b + 3ab² - b³

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8
Q

common mistakes: (x^a)(y^b) =/= xy^a+b

A

Rule only applies when terms are the same.

eg: (x^a) (x^b) = x^a+b

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9
Q

common mistakes: (x+y)^a =/= x^a + y^a

A

Must multiply out the brackets

eg: (x+y)^2 = (x+y)(x+y) = x^2 + 2xy + y^2

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10
Q

common mistakes: (-x)^2 =/= -x^2

A

Look at negative or positive carefully

(-x)^2 = x^2

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11
Q

common mistakes: a/x+y =/= a/x + a/y

A

Denominator must stay the same

eg : x+y/a = x/a + y/a

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12
Q

Simultaneous Equations

A

set of equations that are related

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13
Q

Simultaneous Equations Substitution Method

A
  1. Arrange one of the equations to x = or y =
  2. Substitute that variable in the other equation
  3. Solve
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14
Q

Simultaneous Equations Elimination Method

A
  1. Multiply or divide one equation so they both share the same number one variable (eg: 4x or 5y)
  2. Subtract one equation from the other leaving only the other variable
  3. Solve
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15
Q

Quadratic Equation

A

ax^2 + bx + c =0

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16
Q

Quadratic Formula

A

x = -b±√(b²-4ac) / 2a

± indicates 2 solutions 1 negative 1 positive (notice this on quadratic graphs)

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17
Q

x = -b±√(b²-4ac) / 2a

IF (b²-4ac) = negative number

A

x is not a real number - no real solutions to equation

Square roots of negative numbers are not defined

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18
Q

Solving Quadratics With Factoring

A
  1. ax^2 + bx +c = 0
  2. Factor in to brackets eg: (2x+3)(x-2) = 0
  3. 1 bracket MUST therefore = 0
    SO x -2 = 0 or 2x +3 = 0
  4. Solve - Solutions are x = 2 or -1.5
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19
Q

If you multiply an inequality by a negative number….

A

Direction of inequality symbol is reversed. New inequality is equivalent to the original

20
Q

F(x) - Value of a function when x =

A

eg:

F(x) = 2x +1
F(1) = 2(1) +1 = 3
F(2) = 2(2) +1 = 5
21
Q

Domain of a function - Set of permissible inputs

A

eg: -2< x < 2

22
Q

Simple Annual Interest -

A

Value (V) = P(1+(rt/100))

P = initial investment
r = interest rate 
t = total time invested
23
Q

Annual Compound Interest -

A

V = P(1+ (r/100))^t

P = initial investment
r = interest rate 
t = total time invested
24
Q

Compound Interest paid n times/year

A

V = P(1+ (r/100n))^nt

P = initial investment
r = interest rate 
t = total time invested
25
Q

‘Reflection about the y axis’

A

opposite side of y axis

26
Q

‘Reflection about the x axis’

A

opposite side of y axis

27
Q

‘Reflection about the origin’

A

opposite side of orgin (diagonally)

28
Q

Calculate the distance between 2 points (coordinates)

A
  1. Draw a line between points Q and R
  2. Draw a horizontal line from one point and a vertical line from another point until the 2 lines intersect.
  3. you know the lengths of these 2 new lines and the point at which they intersect is a right angle so use Pythagoras theorem (a² + b² = c²) to find length of QR
29
Q

Slope of horizontal line =

A

Slope of horizontal line = 0

eg: m = y2 - y1/ x2 - x1
= 2 - 2 / 3 - 4
= 0 / 1
= 0

30
Q

Slope of Vertical Line =

A

Slope of Vertical Line = Undefined

eg: m = y2 - y1/ x2 - x1
= 3 -4 / 2 - 2
= 1 / 0
= undefined

31
Q

2 lines are parallel if:

A

Their Slopes are equal

32
Q

2 lines are perpendicular if:

A

Their slopes are are negative reciprocals (flip, make negative) of each other

eg: y = 2x + 1 perpendicular to y = -1/2x + 1

33
Q

2 system equations on same graph

A

solution is where the 2 lines intersect eachother

34
Q

Linear inequality example - y > 2x -1

A
  1. Draw the line y = 2x -1

2. Y can be any value above this line

35
Q

The x intercepts of quadratic graph =

A

solutions for the formula = ax^2 + bx +c = 0

36
Q

Quadratic Equation and its graph

A

If a = positive - U shape

If a = negative - n shape

37
Q

(x-a)²+(y-b)² = r²

A

Circle
Centre Point = (a,b)
r = raidus

38
Q

When lines intersect

A

These are the points where the 2 equations are equal each other eg: f(x) = f(g)

Use this to help solve with other algebra methods

39
Q

Find the 2 points a line intesects a quadratic

A

eg: f(x) = x² , f(g) = 2x +1
1. These are the points where the 2 equations are equal each other

SO: f(x) = f(g)
x² = 2x +1

  1. Simplify to get: x² - 2x + 1 = 0
  2. Solve with quadratic formula or factoring to find the 2 x values
  3. substitute found x values to find corresponding y values
40
Q

Find the 2 points a line intesects a quadratic

A

eg: f(x) = x² , f(g) = 2x +1
1. These are the points where the 2 equations are equal each other

SO: f(x) = f(g)
x² = 2x +1

  1. Simplify to get: x² - 2x + 1 = 0
  2. Solve with quadratic formula or factoring to find the 2 x values
  3. substitute found x values to find corresponding y values
41
Q

graph of h(x) + c

A

shift graph of h(x) UP c units

42
Q

graph of h(x) - c

A

shift graph of h(x) DOWN c units

43
Q

graph of h(x + c)

A

shift graph of h(x) LEFT c units

44
Q

graph of h(x - c)

A

shift graph of h(x) RIGHT c units

45
Q

graph of ch(x) when c>1

A

STRETCH graph of h(x) by c units

46
Q

graph of ch(x) when 0

A

SHRINK graph of h(x) by c units

GRE MATHS (7 decks)

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