Lesson 13: More Factoring Strategies for Quadratic Equations & Expressions

Transcription

1 : More Factoring Strategies for Quadratic Equations & Expressions Opening Exploration Looking for Signs In the last lesson, we focused on quadratic equations where all the terms were positive. Juan s examples had one with mixed signs though. Let s see what happens when the signs are positive, negative or a mixture of both. Use the Factor Bank to help you factor each exercise below. 1. y = x 2 + 5x y = x 2 5x + 6 Factor Bank (-1) (-6) 1 (-6) (-2) (-3) 2 (-3) 3. y = x 2 5x 6 4. y = x 2 + 5x 6 Discussion 5. What patterns do you notice in these four exercises? What generalizations can we make? Unit 11: More with Quadratics Factored Form S.93

2 Create a Factor Bank for the next four problems. Then factor each one. 6. y = x 2 + 9x y = x 2 9x + 8 Factor Bank y = x 2 7x 8 9. y = x 2 + 7x 8 Discussion 10. Did your patterns from Exercise 5 hold for these problems? 11. If the last term (the c value) is positive, what must be true about the two factors you use for the middle term (bx)? 12. If the last term (the c value) is negative, what must be true about the two factors you use for the middle term (bx)? Unit 11: More with Quadratics Factored Form S.94

3 Practice Factor the following quadratic expressions xx xx xx 2 + 5xx x 2 12x x 2 21x xx 2 xx xx 2 + 7xx x 2 2x x 2 + 2x How can you check that your factorization is correct? Unit 11: More with Quadratics Factored Form S.95

4 Exploration A Special Case 22. Consuela ran across the quadratic equation y = 4x 2 16 and wondered how it could be factored. She rewrote it as y = 4x 2 + 0x 16. A. Use the patterns you ve discovered in this lesson and the last lesson to factor this quadratic function. B. What are the key features of the parabola s graph? x-intercepts:, vertex: (, ) C. Graph the quadratic in the grid at the right. y-intercept: Factor the following quadratic functions. Use Consuela s idea of a 0 middle term if necessary. Look for patterns as you go. 23. y = x y = x What is the generic rule for factoring a quadratic in the form a 2 b 2? Unit 11: More with Quadratics Factored Form S.96

5 Homework Problem Set 1. Equations of the form a 2 b 2 are called the difference of perfect squares. Why do you think they have that name? Factor the following examples of the difference of perfect squares. Notice that these are not written as quadratic functions. They are simply expressions we could not graph them or tell their key features. 2. tt xx h 2 36kk bb 2 6. xx xx 6 25 Challenge: Factor each of the following differences of squares completely: 8. 9yy 2 100zz 2 9. aa 4 bb Challenge rr 4 16ss 4 (Hint: This one factors twice.) Unit 11: More with Quadratics Factored Form S.97

6 11. For each of the following, factor out the greatest common factor. a. 6yy b. 27yy yy c. 21bb 15aa d. 14cc 2 + 2cc e. 3xx The measure of a side of a square is xx units. A new square is formed with each side 6 units longer than the original square s side. Write an expression to represent the area of the new square. (Hint: Draw the new square and count the squares and rectangles.) Original Square xx Unit 11: More with Quadratics Factored Form S.98

7 13. In the accompanying diagram, the width of the inner rectangle is represented by xx 3 and the length by xx + 3. The width of the outer rectangle is represented by 3xx 4 and the length by 3xx + 4. a. Write an expression to represent the area of the larger rectangle. xx 3 xx + 3 3xx 4 3xx + 4 b. Write an expression to represent the area of the smaller rectangle. MIXED REVIEW Factor completely xx 2 25xx 15. 9xx xx 2 30xx xx 2 + 7xx xx 2 + 7xx xx xx xx xx xx 2 + 4xx + 1 Unit 11: More with Quadratics Factored Form S.99

8 22. The area of the rectangle at the right is represented by the expression 18xx xx + 2 square units. Write two expressions to represent the dimensions, if the length is known to be twice the width. 1111xx xx Two mathematicians are neighbors. Each owns a separate rectangular plot of land that shares a boundary and has the same dimensions. They agree that each has an area of 2xx 2 + 3xx + 1 square units. One mathematician sells his plot to the other. The other wants to put a fence around the perimeter of his new combined plot of land. How many linear units of fencing does he need? Write your answer as an expression in xx. Note: This question has two correct approaches and two different correct solutions. Can you find them both? Unit 11: More with Quadratics Factored Form S.100