1. ALGEBRA – LESSON 73 Factoring the Difference of Two Squares, Probability without Replacement Be ready to grade the homework!

2. Warm up!  9 = 3  25 = 5  9 = 7  x 2 = x  y 2 = y  x 2 y 2 = xy  9x 2 = 3x  169x 2 = 13x  36a 2 b 2 = 6ab

3. 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.

4. 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. 25x 2 - 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.

5. Factoring a Difference of 2 Squares Steps 25x 2 - 9 1. Find the square root of each term. 5x and 3 2. The answer will be (1 st – 2 nd )(1 st + 2 nd ) (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.

6. Factoring a Difference of 2 Squares Remember: this is a process problem. Recognize it then follow the steps. 16x 2 – 9y 2 1. Find the square root of each term. 4x and 3y 2. The answer will be (1 st – 2 nd )(1 st + 2 nd ) (4x – 3y)(4x + 3y) * ALWAYS look for a common term to factor first! #1

7. Factoring a Difference of 2 Squares -49 + 16y 2 2. Find the square root of each term. 4y and 7 3. The answer will be (1 st – 2 nd )(1 st + 2 nd ) (4y – 7)(4y + 7) #2 1. Rearrange to make this subtraction. 16y 2 - 49

8. Factoring a Difference of 2 Squares 64a 2 x 2 y 2 – 9a 2 z 2 2. Find the square root of each term. 8xy and 3z 3. The answer will be (1 st – 2 nd )(1 st + 2 nd ). Don’t forget the originally factored term. a 2 (8xy – 3z)(8xy + 3z) #3 1. Factor out a common term – even if it’s a perfect square. a 2 (64x 2 y 2 – 9z 2 )

9. 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 2 nd time.

10. 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? 4747 What is the probability for the white marble? (Remember, the 1 st marble was not put back in the bag.) 3636 Multiply 4 3 7 6 x = 2727 #4

11. 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.

12. 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? 7 13 What is the probability for the black gum drop? (Remember, the 1 st gum drop WAS put back in the bag.) 6 13 Multiply 7 6 13 13 x = 42 169 #5 a. With replacement: What is the probability for the yellow gum drop? 7 13 What is the probability for the black gum drop? (Remember, the 1 st gum drop WAS NOT put back in the bag.) 6 12 Multiply 7 6 13 12 x = 7 26 b. Without replacement:

13. 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 1 st orange marble? 7 18 What is the probability for the 2 nd orange marble? (Remember, the 1 st gum drop WAS put back in the bag.) 7 18 Multiply 7 7 18 18 x = 49 324 #6 b. With replacement: What is the probability for the 1 st orange? 7 18 What is the probability for the 2 nd orange marble? (Remember, the 1 st marble WAS NOT put back in the bag.) 6 17 Multiply 7 6 18 17 x = 7 51 a. Without replacement:

14. Homework: PS 73