Quadratic Review- Algebraic Section
By: Alina Farooq
This review will go over Expanding, Factoring, Solving and Completing the Square
What are Quadratic Equations?
For example: y=2x²+6x+8
This equation is in standard form
Main Parts of the Parabola
The vertex
The x-intercepts
The equation
Expanding
Example
(x+a)(x+b)
=x²+bx+ax+ab
=x²+ a+b(x)+ab
First we have multiplied the terms.We multiplied the right side with x and then with a.
Now lets try an example with numbers
(x+2) (x+3)
STEP 1
=x²+3x+2x+6 -We multiplied the first x to (x+3) which gave us x²+3x. Then we did the same with the 2 to (x+3) which gaves us 2x+6. That gave us the equation of x²+3x+2x+6
STEP 2
=x²+5x+6- We added like terms to simplify which gave us x²+5x+6
Now lets try some examples
=x²+5x+6x+30
=x²+11x=30
b) (2x+3) (4x+20
= 8x²+4x+12x+6
= 8x²+16x+6
c) -2(3x+4)(2x-1)
= (-6x-8)(2x-1)
= -12x²+6x-16x+8
= -12x²-10x+8
FACTORING
There are many methods of factoring which will all lead us to solving for the variables. I will be showing you five methods. Common Factoring, Factoring by Grouping, Product Sum, Decomposition, and Difference of Squares
Common Factoring
Example:
21x²+28xy
= 7x(3x+28y)
7x is the greatest common factor.
To check if your answer is correct you can expand your answer and you should get the same equation as it is in the beginning.
24+16x-8x²
=8(3+2x-x²)
x²+4x
=x(x+4)
Factoring by Grouping
Example:
xy-4y+3x-12
=y(x-4) 3(x-4) We can now see terms they have in common and factor further
=(x-4)(y+3)
Product Sum Method
x²+5x-36
(p=-36) (s=5) You take the b value as the sum, and the c value as the product. Using trial and error you find two numbers that add and multiply to those numbers. The numbers are 6 and -1
=(x+6) (x-1)
Decomposition
4x²-5x+6 We will multiply the a value and the c value
Product= 24 Sum=-5 Now we find two numbers that meet these value
The numbers are -8 and 3
4x²-8x+3x+6 We sub in the numbers and then factor by grouping
4x(x-2) 3(x-2)
(4x+3) (x-2) That leaves us with our final answer
Difference of Squares
If you have an equation where there is squared value a minus and then another squared value that can be easily square rooted you can use this method.
Example
x²-81
=(x+9) (x-9)
49x²-25x²
(7x+5y) (7x-5y)
x^4-1
(x²+1) (x²-1)
(x-1)(x+1)(x²+1)
Solving
Solving is actually quite easy after you have factored an equation. We will learn how to solve with a factored equation and solve using the Quadratic Formula.
(x+2) (x-3)
We want to set this equation to zero so we put it into 0=(x+2)(x-3)
You then see what values would need to be plugged in for x for it to equal to zero
The numbers we are left with are x= -2 and x=3
Those values are the two x intercepts of the parabola
Quadratic Formula
x = [ -b ± √(b2-4ac) ] / 2a
Lets plug x²-x-6=0 into the formula
a= 1 b= -1 c=6
After plugging those numbers into the formula the answers we are left with are x=3 x=-2
Those are the two x intercepts
Be careful if there is a negative under your square root that means that there are no x intercepts
Completing the Square
This method is a multiple step process
1.(-5x²+20x)-2 Block off first 2 terms
2. -5(x²+4x)-2 Factor out the A
3. -5(x²+4x+4-4) -2 Take middle term divide by two and square
4. -5(x²+4x+4) 20-2 Take out negative term
5. -5(x+2)²+18 Factor
In this situation the x is the x value of the vertex and the 18 is the y value
Word Problems
Now lets try solving some word problems with our new knowledge
Example 1
To solve this we must set the height to zero and solve for t(time)
0=-5t²+5t+30
= -5(t²-t+6)
Because this is not factorable we must plug it into the quadratic formula
That leaves us with the answers x=-2 x=3
The answer has to be positive so therefore the ball hits the ground in three seconds
Example 2
To solve this question we must complete the square.
P=162x-81x²
=(-81x²+162)
= -81(x²-2+1-1)
=-81(x²-2+1) +81
=-81(x-1)²+81
We are going to use the y value
The maximum profit the company can make is 81 thousand dollars.
Hopefully you now have a clearer understanding of quadratic equations.
Period 4
Mr. Ly