Key Definitions:
Set Notation: (x a < x < b)
Interval Notation (1,2)
It is standard to agree on the following conventions when graphing solutions on the real number line: To include an endpoint, "bubble it in." To exclude an endpoint, use an "empty bubble".
What is the set of all real numbers greater than -2 and less than or equal to 5?
The answer to this question in interval notation would be (2, 5]. The five has a bracket because it is included in the solution.
Example One:
Factor the following Equation:
4x^3- 8^2
x^2 ( 4x-8) Look for the highest common degree of the variable and take it out
4x^2 (x-2) Look for the greatest common factor of the coefficients and take it out
Example Two: Quadratic Equations
ax2+bx+c is the standard form of a quadratic equation.
Factor the following Quadratic Equation:
x^2+7x+ 10
Think of the all the ways to multiply two numbers to multiply two numbers together to get ten.
10 =
1 x 10 -1 x –10
2 x 5 -2 x –5
Then, think, which combinations gives you a sum of 7?
answer:2 and 5
Therefore:
x^2+7x+ 10 = (x +2)(x+5)
Example Three: Graphing Equations
Graph the equation: x2+6x+8
First factor the equation following the above rules to get:
(x+4)(x+2)
x+4=0 and x+2 = 0
There fore x= -4 and –2, these are the place where the equations hit the x-axis.
The point where the graph hits the y-axis can be found by substituting “0” for x in the original equation, to get a value of 8.
Solving Equations
Solving Equations by Factoring
Lastly, quadratic equations are always parabolas. This equation has a first number of positive one, so the graph opens up. It the first number were negative the graph would open down.
