3.c Quadratic Equations

Question 1

How is a quadratic equation different to a linear equation?

Answer 1

Quadratic equations have a variable raised to the second power!

Question 2

What is necessary for a quadratic equation to be factored?

Answer 2

Must be in the form

ax^2 + bx + c = 0

Question 3

How can you factor this equation?

Answer 3

Find two numbers that add to 9, and multiply to be 8.

Hence,
(x + 1)(x + 8) = 0

which means x = -8 or x = -1

Question 4

What does this mean?

(x + 1)(x + 8) = 0

Answer 4

By the Zero Product Property, it must be true that either (x + 1) or (x + 8) is 0, thus x is either -1 or -8

Question 5

What is “foiling” quadratic expressions?

Answer 5

Essentially the opposite of factoring a quadratic expression

Question 6

What is the FOIL process?

Answer 6

It takes an equation in the form of (x+p) (x+q) = 0 and expresses it as ax^2 + bx + c = 0

Question 7

What does FOIL stand for?

Answer 7

First, Outside, Inside, Last

Question 8

Factoring and FOIL are _________ processes

Answer 8

Factoring and FOIL are reverse processes

Question 9

How many quadratic identities do you have to memorise?

Answer 9

3

Question 10

Quadratic identity #1

Answer 10

Question 11

Quadratic identity #2

Answer 11

Question 12

Quadratic identity #3

Answer 12

Question 13

What is this identity also called?

Answer 13

The difference of squares

Question 14

How can you spot the difference of squares?

Answer 14

When one square value is subtracted from another square value

Question 15

Is this an example of the difference of squares?

Answer 15

YES!

x^2 is the square of x, and 1 is the square of 1

Question 16

Is this an example of the difference of squares?

Answer 16

Yes!

4x^2 is the square of 2x and 100 is the square of 10

Question 17

Is this an example of the difference of squares?

Answer 17

Yes!

x^2y^2 is the square of xy, and 16 is the square of 4

Question 18

Is this an example of the difference of squares?

Answer 18

Question 19

Is this an example of the difference of squares?

Answer 19

Question 20

Is this an example of the difference of squares?

Answer 20

Question 21

Is this an example of the difference of squares?

Answer 21

Question 22

Is this an example of the difference of squares?

Answer 22

Question 23

What does this reduce down to?

Answer 23

1

Question 24

What does this reduce down to?

Answer 24

-1

Question 25

What should you do with this equation?

Answer 25

Need to get rid of the fractions! You can do this by multiplying by the equation by the LCM The LCM in this case is x(x+2)

Question 26

Which formula can you use to solve any quadratic equation?

Answer 26

The quadratic formula (the "midnight" formula)

Question 27

What is the quadratic "midnight" formula?

Answer 27

Question 28

What are important conditions required for the quadratic "midnight" formula?

Answer 28

First of all, the quadratic equation must be in the form ax^2 + bx + c = 0, and a can NOT equal 0

Question 29

What is the discriminant?

Answer 29

The discriminant is b^2 - 4ac (the part inside the square root)

Question 30

Why is the discriminant important?

Answer 30

Because it be used to determine the number of roots

Question 31

If the value of the discriminant is positive...

Answer 31

The number of roots is 2

Question 32

If the value of the discriminant is 0...

Answer 32

The number of roots is 1

Question 33

If the value of the discriminant is negative...

Answer 33

The number of roots is 0

Question 34

How are "Vieta's" Formulas useful?

Answer 34

They can be used to easily determine the sum of product of roots

Question 35

What are the two Vieta Formulas?

Answer 35

Question 36

What is required for the Vieta formula to apply?

Answer 36

The quadratic equation must have exactly two solutions

Question 37

What can you do with this formula?

Answer 37

replace x with sqrt(x)^2, and then you can just use a replacement variable such as y