Algebra Flashcards
ca+cb =
ca+cb = c(a+b)
ca-cb =
ca-cb = c (a - b)
(a + b)² =
(a + b)² = a² + 2ab + b²
(a - b)² =
(a - b)² = a² - 2ab + b2
a² - b² =
a² - b² = (a+b)(a-b)
(a + b)³ =
(a+b)³ = a³ + 3a²b + 3ab² - b³
(a - b)³ =
(a - b)³ = a³ - 3a²b + 3ab² - b³
common mistakes: (x^a)(y^b) =/= xy^a+b
Rule only applies when terms are the same.
eg: (x^a) (x^b) = x^a+b
common mistakes: (x+y)^a =/= x^a + y^a
Must multiply out the brackets
eg: (x+y)^2 = (x+y)(x+y) = x^2 + 2xy + y^2
common mistakes: (-x)^2 =/= -x^2
Look at negative or positive carefully
(-x)^2 = x^2
common mistakes: a/x+y =/= a/x + a/y
Denominator must stay the same
eg : x+y/a = x/a + y/a
Simultaneous Equations
set of equations that are related
Simultaneous Equations Substitution Method
- Arrange one of the equations to x = or y =
- Substitute that variable in the other equation
- Solve
Simultaneous Equations Elimination Method
- Multiply or divide one equation so they both share the same number one variable (eg: 4x or 5y)
- Subtract one equation from the other leaving only the other variable
- Solve
Quadratic Equation
ax^2 + bx + c =0
Quadratic Formula
x = -b±√(b²-4ac) / 2a
± indicates 2 solutions 1 negative 1 positive (notice this on quadratic graphs)
x = -b±√(b²-4ac) / 2a
IF (b²-4ac) = negative number
x is not a real number - no real solutions to equation
Square roots of negative numbers are not defined
Solving Quadratics With Factoring
- ax^2 + bx +c = 0
- Factor in to brackets eg: (2x+3)(x-2) = 0
- 1 bracket MUST therefore = 0
SO x -2 = 0 or 2x +3 = 0 - Solve - Solutions are x = 2 or -1.5
If you multiply an inequality by a negative number….
Direction of inequality symbol is reversed. New inequality is equivalent to the original
F(x) - Value of a function when x =
eg:
F(x) = 2x +1 F(1) = 2(1) +1 = 3 F(2) = 2(2) +1 = 5
Domain of a function - Set of permissible inputs
eg: -2< x < 2
Simple Annual Interest -
Value (V) = P(1+(rt/100))
P = initial investment r = interest rate t = total time invested
Annual Compound Interest -
V = P(1+ (r/100))^t
P = initial investment r = interest rate t = total time invested
Compound Interest paid n times/year
V = P(1+ (r/100n))^nt
P = initial investment r = interest rate t = total time invested
‘Reflection about the y axis’
opposite side of y axis
‘Reflection about the x axis’
opposite side of y axis
‘Reflection about the origin’
opposite side of orgin (diagonally)
Calculate the distance between 2 points (coordinates)
- Draw a line between points Q and R
- Draw a horizontal line from one point and a vertical line from another point until the 2 lines intersect.
- 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
Slope of horizontal line =
Slope of horizontal line = 0
eg: m = y2 - y1/ x2 - x1
= 2 - 2 / 3 - 4
= 0 / 1
= 0
Slope of Vertical Line =
Slope of Vertical Line = Undefined
eg: m = y2 - y1/ x2 - x1
= 3 -4 / 2 - 2
= 1 / 0
= undefined
2 lines are parallel if:
Their Slopes are equal
2 lines are perpendicular if:
Their slopes are are negative reciprocals (flip, make negative) of each other
eg: y = 2x + 1 perpendicular to y = -1/2x + 1
2 system equations on same graph
solution is where the 2 lines intersect eachother
Linear inequality example - y > 2x -1
- Draw the line y = 2x -1
2. Y can be any value above this line
The x intercepts of quadratic graph =
solutions for the formula = ax^2 + bx +c = 0
Quadratic Equation and its graph
If a = positive - U shape
If a = negative - n shape
(x-a)²+(y-b)² = r²
Circle
Centre Point = (a,b)
r = raidus
When lines intersect
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
Find the 2 points a line intesects a quadratic
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
- Simplify to get: x² - 2x + 1 = 0
- Solve with quadratic formula or factoring to find the 2 x values
- substitute found x values to find corresponding y values
Find the 2 points a line intesects a quadratic
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
- Simplify to get: x² - 2x + 1 = 0
- Solve with quadratic formula or factoring to find the 2 x values
- substitute found x values to find corresponding y values
graph of h(x) + c
shift graph of h(x) UP c units
graph of h(x) - c
shift graph of h(x) DOWN c units
graph of h(x + c)
shift graph of h(x) LEFT c units
graph of h(x - c)
shift graph of h(x) RIGHT c units
graph of ch(x) when c>1
STRETCH graph of h(x) by c units
graph of ch(x) when 0
SHRINK graph of h(x) by c units
