Quadratic Review- Algebraic Section

By: Alina Farooq

This review will go over Expanding, Factoring, Solving and Completing the Square

What are Quadratic Equations?

Quadratic equations are equations with the degree two. That means that the highest exponent has the value of two. A quadratic forms a parabola.


For example: y=2x²+6x+8

This equation is in standard form

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Main Parts of the Parabola

To find the parts of the parabola we must know how to factor, expand, solve, and, complete the square. These concepts will give us:

Expanding


The first thing we will be learning is Expanding. Essentially we are going to be multiplying binomials and then simplifying.



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

a) (x+6) (x+5)

=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

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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

Factoring is the opposite of expanding. If every term of a polynomial is divisible by the same constant the constant is called a common factor, With common factoring you take out the greatest common factor

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

For factoring by grouping you factor two separate pairs, and then factor those pairs

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

When you cannot find a greatest common factor and the a value is not greater than one you can use this 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

This method is for when there is no greatest common factor and the a value is over 1


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

Another method of factoring is 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.

After you are done factoring an equation you are left with terms like

(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

When you have an equation that is not factorable like x²-x-6=0 you solve using the quadratic formula which is

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

Using the Quadratic Formula

Completing the Square

The last method you will be learning is Completing the square. This method puts the standard form equation ax²+bx+c into vertex form a(x-h)²+k


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

❤² How to Solve By Completing the Square (mathbff)

Word Problems

Now lets try solving some word problems with our new knowledge

Example 1

A baseball is thrown from the top of a building and falls to the gorund below. Its path approximated by the relation h=-5t²+5t+30, where h is the height above the ground in meters and t is the elapsed time in seconds. When will the ball hit the ground?


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

A small company that manufactures snowboards uses the relation P=162x-81x² to model the profit. In the model x represents the number of snowboards in thousands , and P represents the profit in thousands of dollars. What is the maximum profit the company can earn.


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.

Alina Farooq

Period 4

Mr. Ly