- One variable is related to another by a function involving constant terms and terms of 1st order or higher.
- A polynomial with a degree of 2 is Quadratic Relation.
- For constant increments of the independent variable, a relation is quadratic if the second differences of the dependent variable are constant.
- A graph that is either concave upwards or concave downwards
- When the rate of change is constant it is a linear relation
- When the rate of change is not constant then its not a linear relation
- Factoring is finding what to multiply to get an expression
- To factor a number means breaking it into an equation where a number can be multiplied to get the original
There are many different types of factoring I will explain all.
Firstly we have - Common Factoring
- If all the terms in the equation are divisible by the same number or variable then that term is called the common factor
- A polynomial is not completely factored if the GCF (greatest common factor) is not factored out.
- Expanding and factoring are 2 completely different things a(b+c) <= (factoring) ab+bc <= (Expanding)
=> (factored Fully because 25 is the GCF)
=> (It is NOT factored fully as 5 and 10 still have a common factor)
=> x^2(x+2) +8(x+2)
- Simple trinomial equations are in the form of, x^2+bx+c
- In solving trinomials, the brackets have to be expanded (Multiply each term in the first bracket to each term in the second bracket).
- Equations in the form of x^2+bx+c are called Quadratic Equations
Expand=> x^2+bx+c , (x+2) (x+1)
- (x+2) (x+1)
- "c" value is the product. Find out 2 numbers that add up to the "b" value and when multiplied the answer is the "c" value.
- x^2 +7x -30 [Product=-30 , Sum=7]
= (x+10) (x-3)
- Factor by Decomposition.
- Complex Trinomial are in the form of, Ax^2 +Bx +C.
- The "C" value is multiplied by the A value, only the coefficient.
- Determine the factors of the product which also add up to the "B" value.
Factor by Decomposition: Ax^2 +Bx +C
- 2x^2 +3x -5 [Product: A*C= -5*2=-10 , Sum=3]
=2x^2 +5x-2x -5 [Decompose the Middle term]
- 5x^2 +10x+2x+4
FACTORING DIFFERENCE OF SQUARES
FACTORING DIFFERENCE OF SQUARES
- The "a" value is squared (x^2)
- In the form of, (a+b) (a-b)
- Square-root of "b" value
- (x+2) (x-2)
=x^2 -2x+2x -4
=x^2 - 4
- x^2 -16
= (x^2 - 4)
4x^2 - 22
=2(2x^2 - 11)
COMPLETING THE SQUARE
- Used to change Standard form equations to Vertex form
STEPS TO COMPLETE THE SQUARES:
- Block off the first two terms
- Factor out the A value only
- Divide the middle term by 2 and square it (b/2)^2
- Add and subtract the squared value answer
- Take out the negative squared value, Multiply the "a" value by the negative squared value when taking it out of the bracket
- Write within brackets x + the square root of the number and square the entire bracket and solve what is outside the brackets
- x^2 + 6x - 2
=(x^2 + 6x) - 2
=(x^2 + 6x + 9 - 9) -2
=(x^2 + 6x + 9) -9 - 2
=(x+3)^2 - 11
- In the form of, ax^2 + bx + c = 0
- Formula is used to identify the "x" value
- Quadratic formula is used when the equation cannot be factored
- Quadratic formula give 2 values of "x"
- Quadratic Formula:
x = -b ± √b^2 - 4ac /2a
- x^2 - x + 8 = 0
x= -b ± √b^2 - 4ac /2a
x= 4x^2 + 33x + 8 = 0
x= -33 ± √33^2 - 4(4)(8) / 2(4)
x= -33 ± √961 / 8
x= -33 ± 31 / 8
x= -1/4 x= -8