Quadratics Review Assignment-Unit 4
By: Zoya Naseem
Linear and Non-Liner relations
Degree
5x has a degree of 1
x^3 has a degree of 3
2x^2 has a degree of 2
5x^2-3x+2 has a degree of 2
Quadratic Formula
ax^2+bx+c
The video below will explain more and give examples on quadratic formula.
Multiplying Binomials
Steps to solve:---------->(x+a)(x+b)
1st step is to multiply the first bracket with the second bracket
(x)(x)=x^2
(x)(b)=bx
2nd step is to multiply "a" with the rest in the other bracket
(a)(x)=ax
(a)(b)=ab
3rd step is to expand it
=x^2+bx+ax+ab
=x^2+ax+bx+ab
=x^2+(a+b)x+ab
How I got this:
x^2 because (x)(x) gives x^2
(a+b)x because you add the like terms
ab because a value times b value gives us the number
Example
(x+2)(x+3)
=x^2+3x+2x+6
=x^2+5x+6
Common Factoring
xy+xz=x(y+z)-----------------------same as------------------------> 5x+35=5(x+7)
I have made 5x+35 into 5(x+7) because 5 goes into 35, 7 times.
Factor by Grouping
Trinomial= (Binomial)(Binomial) ----------------> Quadratic Expression
x^2+15x+56
Sum:15 and Product:56 ---------------->(7)(8)=56
--------------------------------------------------->7+8=15
(x+7)(x+8)
The video below will give more explanation and examples about factor by grouping
Factoring Simple Trinomials
=x^2+7x+6
Your b (what it is) should be the sum of two numbers and those 2 numbers should be the product of c value. And in this case the two numbers are.....
Sum: 7 Product: 6----------->(6)(1)=6
-------------------------------------> 6+1=7
=x^2+6x+1x+6
=x(x+6)+1(x+6)
=(x+1)(x+6)
Factoring Complex Trinomials
ax^2+bx+c------------------> Expression
Example:
= 8x^2+22x+15
=(8)(15)=120
=8x^2+22x+120
Sum: 22 and Product:120----------->(10)(12)=120
----------------------------------------------->10+12=22
=8x^2+10x+12x+15
=2x(4x+5)+3(4x+5)
=(4x+5)(2x+3)
Solving Quadratics by Factoring
Example 1:
2x+5=0
2x/2=-5/2
x=-5/2
Example 2:
x^2-5x-14=0
Sum: -5 and Product: -14----------->(-7)(2)=-14
-----------------------------------------------> -7+2=-5
x^2-7x+2x-14=0
x(x-7)+2(x-7)
(x+2)(x-7)
x=-2, x=7
Note: any numbers that move across the equal sign, changes into the opposite sign
Completing the Square
y=(x-h)^2+k which is known as completing the square/vertex form.
Example: ---------------------->y=x^2+8x-3
- Block the first two terms ------------> y=(x^2+8x)-3
- Factor out the "a"
- divide the middle term by two, then square it -------------------------------------->(8/2)^2=16 y=(x^2+8x+16-16)-3
- take out the negative number inside the bracket-----------------------> y=(x^2+8x+16)-16-3
- y=(x+4)^2-19 <---------------------------This is known as vertex form
Note: If the "a" value had a number then on the 4th step, multiply the "a" value with the negative number which is inside the bracket, then take it out of the bracket.
Challenging Questions
Question 1:
2x+1=√2x+1(2x+1)^2=(√2x+1)
(2x+1)^2=2x+1
[(2x+1)(2x+1)]-(2x+1)=0
[4x^2+2x+2x+1]-(2x-1)=0
(4x^2+4x+1)-2x-1=0
4x^2+2x+0=0
2x(2x+1)+0=0
2x=0, 2x+1=0
x=0,x=-1/2
Question 2:
n^2-n(n-1)/2=0
(2)n^2=n(n-1) <----- cancelling the 2 out to get rid of the 2 that was being divided by n(n-1)
2n^2=n^2-n
2n^2-n^2+n=0
n(n+1)=0
n=0, n+1=0
n=0, n=-1
Question 3:
(x-3)x=4
x^2-3x-4=0
sum: -3 and Product: -4-------->(-4)(1)=-4
------------------------------------------>-4+1=-3
x^2-4x+1x-4=0
x(x-4)+1(x-4)=0
x+1=0, x-4=0
x=-1, x=4