Algebra

Question 1

ca+cb =

Answer 1

ca+cb = c(a+b)

Question 2

ca-cb =

Answer 2

ca-cb = c (a - b)

Question 3

(a + b)² =

Answer 3

(a + b)² = a² + 2ab + b²

Question 4

(a - b)² =

Answer 4

(a - b)² = a² - 2ab + b2

Question 5

a² - b² =

Answer 5

a² - b² = (a+b)(a-b)

Question 6

(a + b)³ =

Answer 6

(a+b)³ = a³ + 3a²b + 3ab² - b³

Question 7

(a - b)³ =

Answer 7

(a - b)³ = a³ - 3a²b + 3ab² - b³

Question 8

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

Answer 8

Rule only applies when terms are the same.

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

Question 9

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

Answer 9

Must multiply out the brackets

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

Question 10

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

Answer 10

Look at negative or positive carefully

(-x)^2 = x^2

Question 11

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

Answer 11

Denominator must stay the same

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

Question 12

Simultaneous Equations

Answer 12

set of equations that are related

Question 13

Simultaneous Equations Substitution Method

Answer 13

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

Question 14

Simultaneous Equations Elimination Method

Answer 14

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

Question 15

Quadratic Equation

Answer 15

ax^2 + bx + c =0

Question 16

Quadratic Formula

Answer 16

x = -b±√(b²-4ac) / 2a

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

Question 17

x = -b±√(b²-4ac) / 2a

IF (b²-4ac) = negative number

Answer 17

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

Square roots of negative numbers are not defined

Question 18

Solving Quadratics With Factoring

Answer 18

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

Question 19

If you multiply an inequality by a negative number….

Answer 19

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

Question 20

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

Answer 20

eg:

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

Question 21

Domain of a function - Set of permissible inputs

Answer 21

eg: -2< x < 2

Question 22

Simple Annual Interest -

Answer 22

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

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

Question 23

Annual Compound Interest -

Answer 23

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

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

Question 24

Compound Interest paid n times/year

Answer 24

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

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

Question 25

'Reflection about the y axis'

Answer 25

opposite side of y axis

Question 26

'Reflection about the x axis'

Answer 26

opposite side of y axis

Question 27

'Reflection about the origin'

Answer 27

opposite side of orgin (diagonally)

Question 28

Calculate the distance between 2 points (coordinates)

Answer 28

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

Question 29

Slope of horizontal line =

Answer 29

Slope of horizontal line = 0 eg: m = y2 - y1/ x2 - x1 = 2 - 2 / 3 - 4 = 0 / 1 = 0

Question 30

Slope of Vertical Line =

Answer 30

Slope of Vertical Line = Undefined eg: m = y2 - y1/ x2 - x1 = 3 -4 / 2 - 2 = 1 / 0 = undefined

Question 31

2 lines are parallel if:

Answer 31

Their Slopes are equal

Question 32

2 lines are perpendicular if:

Answer 32

Their slopes are are negative reciprocals (flip, make negative) of each other eg: y = 2x + 1 perpendicular to y = -1/2x + 1

Question 33

2 system equations on same graph

Answer 33

solution is where the 2 lines intersect eachother

Question 34

Linear inequality example - y > 2x -1

Answer 34

1. Draw the line y = 2x -1 | 2. Y can be any value above this line

Question 35

The x intercepts of quadratic graph =

Answer 35

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

Question 36

Quadratic Equation and its graph

Answer 36

If a = positive - U shape | If a = negative - n shape

Question 37

(x-a)²+(y-b)² = r²

Answer 37

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

Question 38

When lines intersect

Answer 38

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

Question 39

Find the 2 points a line intesects a quadratic

Answer 39

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 2. Simplify to get: x² - 2x + 1 = 0 3. Solve with quadratic formula or factoring to find the 2 x values 4. substitute found x values to find corresponding y values

Question 40

Find the 2 points a line intesects a quadratic

Answer 40

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 2. Simplify to get: x² - 2x + 1 = 0 3. Solve with quadratic formula or factoring to find the 2 x values 4. substitute found x values to find corresponding y values

Question 41

graph of h(x) + c

Answer 41

shift graph of h(x) UP c units

Question 42

graph of h(x) - c

Answer 42

shift graph of h(x) DOWN c units

Question 43

graph of h(x + c)

Answer 43

shift graph of h(x) LEFT c units

Question 44

graph of h(x - c)

Answer 44

shift graph of h(x) RIGHT c units

Question 45

graph of ch(x) when c>1

Answer 45

STRETCH graph of h(x) by c units

Question 46

graph of ch(x) when 0

Answer 46

SHRINK graph of h(x) by c units