Quadratic Expressions and Equations Unit 8 Quadratic Expressions and Equations EQ: How do you use addition, subtraction, multiplication, and factoring of polynomials in order to simplify rational expressions? Splash Screen

Differences of Squares Lesson 7 Differences of Squares Essential Question: How do you factor binomials that are the difference of squares and use factored form to solve equations? Splash Screen

5 minute check on previous lesson. Do the first 5 problems! Lesson Menu

Factor m2 – 13m + 36. A. (m – 4)(m – 9) B. (m + 4)(m + 9) Review over lesson 6 Factor m2 – 13m + 36. A. (m – 4)(m – 9) B. (m + 4)(m + 9) C. (m + 6)(m – 6) D. (m + 6)2 5-Minute Check 1

Factor –24 – 5x + x2. A. (x + 3)(x + 8) B. (x – 3)(x – 8) Review over lesson 6 Factor –24 – 5x + x2. A. (x + 3)(x + 8) B. (x – 3)(x – 8) C. (x + 8)(x – 3) D. (x + 3)(x – 8) 5-Minute Check 2

Solve y2 – 8y – 20 = 0. A. {–4, 3} B. {3, 6} C. {–2, 10} D. {1, 8} Review over lesson 6 Solve y2 – 8y – 20 = 0. A. {–4, 3} B. {3, 6} C. {–2, 10} D. {1, 8} 5-Minute Check 3

Solve x2 + 8x = –12. A. {–8, –4} B. {–6, –2} C. {–4, 4} D. {2, 3} Review over lesson 6 Solve x2 + 8x = –12. A. {–8, –4} B. {–6, –2} C. {–4, 4} D. {2, 3} 5-Minute Check 4

Which shows the factors of p8 – 8p4 – 84? Review over lesson 6 Which shows the factors of p8 – 8p4 – 84? A. (p4 – 14)(p4 + 6) B. (p4 + 7)(p2 – 12) C. (p4 – 21)(p4 – 4) D. (p4 – 2)(p2 + 24) 5-Minute Check 6

Differences of Squares Lesson 7 Differences of Squares Essential Question: How do you factor binomials that are the difference of squares and use factored form to solve equations? Splash Screen

You factored trinomials into two binomials. Factor binomials that are the difference of squares. Use the difference of squares to solve equations. EQ: How do you factor binomials that are the difference of squares and use factored form to solve equations? Then/Now

difference of two squares EQ: How do you factor binomials that are the difference of squares and use factored form to solve equations? Vocabulary

From lesson 4, do you remember the pattern for a product of a Sum and a Difference? (a + b)(a - b) = a2 - b2 Concept

a2 – b2 = (a + b)(a – b) a2 – b2 = (a – b)(a + b) Concept

m2 – 64 = m2 – 82 Write in the form a2 – b2. Factor Differences of Squares A. Factor m2 – 64. m2 – 64 = m2 – 82 Write in the form a2 – b2. = (m + 8)(m – 8) Factor the difference of squares. Answer: (m + 8)(m – 8) Example 1

16y2 – 81z2 = (4y)2 – (9z)2 Write in the form a2 – b2. Factor Differences of Squares B. Factor 16y2 – 81z2. 16y2 – 81z2 = (4y)2 – (9z)2 Write in the form a2 – b2. = (4y + 9z)(4y – 9z) Factor the difference of squares. Answer: (4y + 9z)(4y – 9z) Example 1

C. Factor y2 – 121. = (y + 11)(y – 11) D. Factor 4m2 – 49n2. Factor Differences of Squares C. Factor y2 – 121. = (y + 11)(y – 11) D. Factor 4m2 – 49n2. = (2m + 7n)(2m – 7n) Example 1

3b3 – 27b = 3b(b2 – 9) The GCF of 3b2 and 27b is 3b. Factor Differences of Squares E. Factor 3b3 – 27b. If the terms of a binomial have a common factor, the GCF should be factored out first before trying to apply any other factoring technique. 3b3 – 27b = 3b(b2 – 9) The GCF of 3b2 and 27b is 3b. = 3b[(b)2 – (3)2] Write in the form a2 – b2. = 3b(b + 3)(b – 3) Factor the difference of squares. Answer: 3b(b + 3)(b – 3) Example 1

F. Factor the binomial b2 – 9. A. (b + 3)(b + 3) B. (b – 3)(b + 1) C. (b + 3)(b – 3) D. (b – 3)(b – 3) Example 1

G. Factor the binomial 25a2 – 36b2. A. (5a + 6b)(5a – 6b) B. (5a + 6b)2 C. (5a – 6b)2 D. 25(a2 – 36b2) Example 1

H. Factor 5x3 – 20x. A. 5x(x2 – 4) B. (5x2 + 10x)(x – 2) C. (x + 2)(5x2 – 10x) D. 5x(x + 2)(x – 2) Example 1

Assignment Do Worksheet #1 to #9 EQ: How do you factor binomials that are the difference of squares and use factored form to solve equations? End of the Lesson

y4 – 625 = [(y2)2 – 252] Write y4 – 625 in a2 – b2 form. Apply a Technique More than Once A. Factor y4 – 625. y4 – 625 = [(y2)2 – 252] Write y4 – 625 in a2 – b2 form. = (y2 + 25)(y2 – 25) Factor the difference of squares. = (y2 + 25)(y2 – 52) Write y2 – 25 in a2 – b2 form. = (y2 + 25)(y + 5)(y – 5) Factor the difference of squares. Answer: (y2 + 25)(y + 5)(y – 5) Example 2

256 – n4 = 162 – (n2)2 Write 256 – n4 in a2 – b2 form. Apply a Technique More than Once B. Factor 256 – n4. 256 – n4 = 162 – (n2)2 Write 256 – n4 in a2 – b2 form. = (16 + n2)(16 – n2) Factor the difference of squares. = (16 + n2)(42 – n2) Write 16 – n2 in a2 – b2 form. = (16 + n2)(4 – n)(4 + n) Factor the difference of squares. Answer: (16 + n2)(4 – n)(4 + n) Example 2

C. Factor y4 – 16. A. (y2 + 4)(y2 – 4) B. (y + 2)(y + 2)(y + 2)(y – 2) C. (y + 2)(y + 2)(y + 2)(y + 2) D. (y2 + 4)(y + 2)(y – 2) Example 2

D. Factor 81 – d4. A. (9 + d)(9 – d) B. (3 + d)(3 – d)(3 + d)(3 – d) C. (9 + d2)(9 – d2) D. (9 + d2)(3 + d)(3 – d) Example 2

9x5 – 36x = 9x(x4 – 4) Factor out the GCF. Apply Different Techniques A. Factor 9x5 – 36x. 9x5 – 36x = 9x(x4 – 4) Factor out the GCF. = 9x [(x2)2 – 22] Write x2 – 4 in a2 – b2 form. = 9x (x2 + 2)(x2 – 2) Factor the difference of squares. Answer: 9x(x2 + 2)(x2 – 2) Example 3

6x3 + 30x2 – 24x – 120 Original polynomial Apply Different Techniques B. Factor 6x3 + 30x2 – 24x – 120. 6x3 + 30x2 – 24x – 120 Original polynomial = 6(x3 + 5x2 – 4x – 20) Factor out the GCF. = 6[(x3 + 5x2) + (– 4x – 20)] Group terms with common factors. = 6[x2(x + 5) – 4(x + 5)] Factor each grouping. = 6(x + 5)(x2 – 4) x + 5 is the common factor. = 6(x + 5) (x + 2)(x – 2) Factor the difference of squares. Answer: 6(x + 5)(x + 2)(x – 2) Example 3

C. Factor 3x5 – 12x. A. 3x(x2 + 3)(x2 – 4) B. 3x(x2 + 2)(x2 – 2) C. 3x(x2 + 2)(x + 2)(x – 2) D. 3x(x4 – 4x) Example 3

D. Factor 5x3 + 25x2 – 45x – 225. A. 5(x2 – 9)(x + 5) B. (5x + 15)(x – 3)(x + 5) C. 5(x + 3)(x – 3)(x + 5) D. (5x + 25)(x + 3)(x – 3) Example 3

Assignment Do Worksheet #10 to #21 EQ: How do you factor binomials that are the difference of squares and use factored form to solve equations? End of the Lesson

4y2 = 81 A. Solve the equation by factoring 4y2 = 81. 4y2 – 81 = 0 2y + 9 = 0 OR 2y – 9 = 0 2y = -9 2y = 9 y = -9/2 y = 9/2 Answer: {- 4.5 , 4.5 } Example 4

75x3 = 147x B. Solve the equation by factoring 75x3 = 147x. 3x (5x + 7)(5x – 7) = 0 3x = 0 OR 5x + 7 = 0 OR 5x – 7 = 0 x = 0 5x = -7 5x = 7 x = -7/5 x = 7/5 Answer: {- 1.4, 0 , 1.4 } Example 4

C. In the equation which is a value of q when y = 0? A B C 0 D Read the Test Item Factor as the difference of squares. Solve the Test Item Original equation Replace y with 0. Example 4

Factor the difference of squares. Write in the form a2 – b2. Factor the difference of squares. or Zero Product Property Solve each equation. Answer: Correct answer is D. Example 4

D. In the equation m2 – 81 = y, which is a value of m when y = 0? B. C. –9 D. 81 Example 4

Assignment Finish the Worksheet. EQ: How do you factor binomials that are the difference of squares and use factored form to solve equations? End of the Lesson