ALGEBRA – LESSON 73Factoring the Difference of Two Squares, Probability without Replacement

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9 = 3 25 = 5 9 = 7

x2 = x y2 = y x2y2 = xy

9x2 = 3x 169x2 = 13x 36a2b2 = 6ab

Factoring a Difference of 2 Squares (not in your notes)

We have learned that factoring means to break into parts that can be multiplied to result in the original equation.

We have learned a couple of strategies for factoring:

1 – We can “unfoil”.

2 – We can look for something that every term has in common.

Today we will learn another strategy that involves very little math. It is simply recognizing the type of equation and then using a formula for plugging in the numbers.

Factoring a Difference of 2 Squares (not in your notes)

We need to be on the lookout for a special kind of factoring problems. If the equation has 2 terms which

are being subtracted and both terms are perfect squares.

25x2 - 9

• 2 terms

•Both terms formed from perfect squares. (numbers AND variables)

•The terms are being subtracted.

•Hint 1: you may have to rearrange it for it to be subtraction.

•Hint 2: you may have to pull out a common factor first.

Factoring a Difference of 2 Squares

Steps

25x2 - 9

1. Find the square root of each term.

5x and 3

2. The answer will be (1st – 2nd)(1st + 2nd)

(5x – 3)(5x + 3)

* ALWAYS look for a common term to factor first!

If the equation has 2 terms and both terms are perfect squares and 1 is positive and 1 is negative, we call them the difference of 2 squares.

Factoring a Difference of 2 Squares

Remember: this is a process problem. Recognize it then follow the steps.

16x2 – 9y2

1. Find the square root of each term.

4x and 3y

2. The answer will be (1st – 2nd)(1st + 2nd)

(4x – 3y)(4x + 3y)

* ALWAYS look for a common term to factor first!

#1

Factoring a Difference of 2 Squares

-49 + 16y2

2. Find the square root of each term.

4y and 7

3. The answer will be (1st – 2nd)(1st + 2nd)

(4y – 7)(4y + 7)

#2

1. Rearrange to make this subtraction.

16y2 - 49

Factoring a Difference of 2 Squares

64a2x2y2 – 9a2z2

2. Find the square root of each term.

8xy and 3z

3. The answer will be (1st – 2nd)(1st + 2nd). Don’t forget the originally factored term.

a2(8xy – 3z)(8xy + 3z)

#3

1. Factor out a common term – even if it’s a perfect square.

a2(64x2y2 – 9z2)

Probability without replacement

We have calculated probability of independent events. But this process will change as soon as we begin to see problems in which

objects ARE NOT replaced. For example – a marble is drawn and then NOT put back in the bag.

We will calculate these probabilities by: multiplying each probability taking into account that there are fewer total

items the 2nd time.

Probability without replacement

A bag contains 3 white marbles and 4 yellow marbles. One marble is drawn at random and not replaced. Then a

second marble is drawn. What is the probability that the first marble is yellow and the second one is white?

What is the probability for the yellow marble?

47

What is the probability for the white marble? (Remember, the 1st marble was not put back in the bag.)

36

Multiply

4 37 6

x = 27

#4

Probability without replacement

Problems will now be asking for you to calculate the probability with replacement and without. Think about why

you are choosing the fractions you are choosing.

Also be aware that sometimes they’ll switch the order that they ask the questions. Notice #5 and #6 on your notes.

Probability without replacement

Martha has a bag that contains 7 yellow gum drops and 6 black gum drops. She randomly draws 2 gum drops, one after the other. What is the probability

that the first gum drop is yellow and the second gum drop is black?

What is the probability for the yellow gum drop?

713

What is the probability for the black gum drop? (Remember,

the 1st gum drop WAS put back in the bag.)

613

Multiply

7 613 13

x = 42169

#5

a. With replacement:

What is the probability for the yellow gum drop?

713

What is the probability for the black gum drop? (Remember,

the 1st gum drop WAS NOT put back in the bag.)

612

Multiply

7 613 12

x = 726

b. Without replacement:

Probability without replacement

Larry has a urn that contains 7 orange marbles and 11 blue marbles. He randomly draws 2 marbles, one after the other. What is the probability that

both marbles are orange?

What is the probability for the 1st orange marble?

718

What is the probability for the 2nd orange marble?

(Remember, the 1st gum drop WAS put back in the bag.)

718

Multiply

7 718 18

x = 49324

#6

b. With replacement:

What is the probability for the 1st orange?

718

What is the probability for the 2nd orange marble?

(Remember, the 1st marble WAS NOT put back in the bag.)

617

Multiply

7 618 17

x = 751

a. Without replacement:

Homework: PS 73