Quadratics
By Jasman S.
Introduction:
Table of contents:
Vertex form
Step pattern
Parts of a parabola
Factored form
Standard form
Converting standard form to vertex form
How 3 form relate
Factoring
Expand and simplify
Common factoring
Factoring simple trinomials
Factoring complex trinomials
Factoring perfect square trinomials
Factoring differences in squares
Quadratic formula
Discriminant
Word problem
Reflection
Basics
What is a parabola?
a graph of quadratic relations that can open downward or upwards which means it could either be positive or negative. meaning it could be minimum or maximum
Vertex Form
a(x – h)2 + k
In order to know the vertex form you need to know the basic steps:
THE FORMULA
THE STEP PATTERN
HOW TO GRAPH IT
PROPER TERMINOLOGY
Step pattern
If you take the "standard" parabola, y = x², which has it's vertex at the origin (0, 0), then:
Starting from the vertex (0,0) as "the first point" ...
OVER 1 (right or left) from the vertex point, UP 1² = 1 from the vertex point
OVER 2 (right or left) from the vertex point, UP 2² = 4 from the vertex point
Parts of a parabola
Factored form
The form of an algebraic expression in which no part of the expression can be made simpler by pulling out a common factor.
In factored form, there are two x intercepts (s&t)
EX: y=-2(x+3)(x+1)
x intercepts @ (-3,0)(-1,0)
(-3+-1)/2
Therefore; the axis of symmetry is -2.
Standard form
a, b and c are known values. a can't be 0.
"x" is the variable or unknown (we don't know it yet
a, b and c are known values. a can't be 0.
"x" is the variable or unknown (we don't know it yet
a, b and c are known values. a can't be 0.
"x" is the variable or unknown (we don't know it yet
Standard form cannot be graphed therefore we convert standard form to vertex form
Converting standard form to vertex form
Since we cannot graph standered form. how do we change it to vertex form?
We complete a perfect square!
Direction of opening
Factoring:
Expand and Simplify:
To expand we get rid of the brackets.
Distributive law:
a(b + c) = ab + ac
We use the distributive law to write expressions such as 5(x + 4) in a form that has no brackets,
5(x + 4)
= 5x + 20
Common factoring
You know how to multiply two brackets together. (expand and simplify)
(a + b)(c + d) = ac + ad + bc + bd
If you have the expression on the right hand side how can you change it back to the two brackets on the left hand side?
When you common factor, you are breaking it up by finding a number or expression/exponant that is common
Example: (2x+6)
there is a common factor of 2
=2(x+2)
Example 2:
Factoring Simple Trinomials x^2+bx+c
Factor:
Example 1.
1)x^2+6x+5
= (x+1)(x+5)
Solve: =x^2+5x+1x+5
x^2+6x+5
Example 2:
x^2+10x+16
=(x+8)(x+2)
Solve= x^2+2x+8x+16
=x^2+10x+16
Example 3:
x^2+7x+12
=(x+3)(x+4)
Solve: x^2+4x+3x+12
=x^2+7x+12
Factoring Complex Trinomials ax^2+bx+c
A complex trinomial is a trinomial that does NOT start with 1.
For example: 3x^2+18x+12
2x^2+11x+15
Look for a common factor, and factor it into 2 sections
Break into 2 factors; guess and check
=(2x+5)(x+3)
you need to find 2 numbers that multiply to 15.
Put the first two terms in brackets and then the last two terms in brackets
Then factor
Finally , put the terms outside the brackets in one separate bracket, and the terms already in brackets combined into one bracket
Factoring perfect square trinomials
Perfect squares start with a squared number and end with a squared number with a number in between (a^2+2ab+b^2)
Example 1:
2x^+8x+4
=(2x+2)(x+2)
then expand to check
2x(2x^2+4x)^2+(2x+4)
=2x^+8x+4
Factoring differences of squares
Difference of squares have nothing in the middle (x^2-36)
Example: x^2-36
=(x+6)(x-6) because the square root of 36 is 6 and when the equation is expanded it goes back to original form indicating that it is the right factors.