Algebra Honors Unit 3 Factoring Quadratics Name Quest Mission 1 Factoring by Greatest Common Factor and Grouping Review Questions 1. Simplify: i(6 4i) 3+3i A. 4i C. 60 + 3 i B. 8 3 + 4i D. 10 3 + 3 i. Solve 4x = - 0 Objectives: SWBAT factor out a Greatest Common Factor from polynomials. SWBAT identify polynomials that are prime. SWBAT factor polynomials by grouping. SWBAT use the zero product property to solve a quadratic. GCF - Prime - Find the greatest common factor. 1. 6 and 15. 16, 4 and 36 3. 3x and 1x Factor out the greatest common factor. 4. 6x 14 5. 4x y 6xy 6. 8x + 9z 7. 18w 3 15w 4 + 5w 5 Factor by Grouping. 8. a(b + 4) + c(b + 4) 9. a(x 3) + x (x 3) 10. 3y 8y 1y + 3 11. ab + bc + a + c
ZERO PRODUCT PROPERTY Algebra: If A and B are expressions and AB = 0, then A = or B =. Example: If (x + 5)(x + ) = 0, then = 0 or = 0. So x = or x = Finding the Z.A.R.S. Zeros~ Answers~ Roots~ Solutions~ Solve the following. 1. (x + 3)(x 5) = 0 13. y(4y 9) = 0 14. a 1a = 0 15. 5z = 30z 16. y y y 5 5 5 0 17. 8x 6x 1x 9 0
Mission Factoring Hard Trinomials (a > 1) Review Questions 1. Write 7i(1+i)+5 3i A. B. 16i 3 9 + 7i 3i as a complex number in standard form. C. 7 3 + 3i D. 7 + 5 3 i. Solve 6x = 1x Objectives: SWBAT factor hard trinomials, trinomials with a leading coefficient greater than 1. SWBAT solve a quadratic equation with a leading coefficient greater than 1. Factor ax + bx + c (using the x-factor and factor by grouping). 1. x 9x 7 ax bx c. 3x 7x 0 3. 5x 9x 0
Find the zeros of y = ax + bx + c (By factoring and using the zero product rule). 4. y x x 1 5. f x x x ( ) 4 5 6 Find the roots of ax + bx + c = 0 (By factoring and using the zero product rule). 6. w 8w 1 7. 3x 18x 3x 4x
Mission 3 Factoring Easy Trinomials (a = 1) and Factoring by Substitution Review Questions 1. Solve: x 4 64 = 0 A. x = ± 8, x = ±8i C. x = ±8, x = ±8i B. x = ±, x = ±i D. x = ±4, x = ±4i. Solve - w = w. Objectives: SWBAT factor easy trinomials, trinomials with a leading coefficient of 1. SWBAT solve a quadratic equation with a leading coefficient of 1. SWBAT factor quadratics using substitution. Factor trinomials in the form x + bx + c (using the x-factor). 1. x + 7x 8. n + 7n +1 3. x 4x 1
Find the roots of ax + bx + c = 0 (By factoring and using the zero product rule). 4. x x 15 = 0 5. x + = 3x 6. Find the zeros of: f(x) = x + 3x 40 7. x + 8x 4 = 0 8. 4k + 14k = 30 9. 0 = 3x + 33x + 36 Factor quadratics using substitution. 10. (x + 1) + 4(x + 1) + 4 11. (x ) + 4(x ) 1
Mission 4 Special Factoring Patterns Review Questions 1. Factor: 4x 4 13x + 9 A. (x 9)(x 4) C. (4x 9) (x 1) B. (x 3)(x + 3)(x + 1)(x 1) D. (4x 3)(x + 3)(x + 1)(x 1). Find the zeros of f(x) = x 9x + 0. Objectives: SWBAT identify different types of factoring problems. SWBAT identify and use special factoring patterns. SWBAT factor the un-factorable. SPECIAL FACTORING PATTERNS Difference of Two Squares: a b = ( )( ) Example: x 4 = Perfect Square Trinomial: a + ab + b = ( ) Example: x + 6x + 9 = Perfect Square Trinomial: a ab + b = ( ) Example: 16x 56x + 49 = Factor with special patterns. 1. x 5. m m + 11 3. d 64 4. x + 1x + 36 5. 4x + 36x + 81 6. 9x 5 7. 5d 110d + 11 8. 4x + 36 9. d 1 Find the roots of the following equations. 10. x + 18 = 9x 11. x 5x + 3 = 0
1. 4x + 36 = 0 13. 4x 3 4x = 4x Solve the following. 14. 4m 0m + 5 = 0 15. x = 50 16. 3y 3 y 1y + 8 = 0 17. x 11 = 0
Review Questions Mission 5 Solve Quadratic Equations by Finding Square Roots 1. Factor the following using imaginary numbers: 9x + 49 A. (3x 7) C. (3x + 7i)(3x 7i) B. ( 3x + 7)( 3x 7) D. (3x + 7i). Solve 3x = 7 3. Simplify 4 and 98 4. Simplify 17 16 Objective: SWBAT solve quadratic equations by using square roots. Square root - Principal square root - Radical - Radicand - Rationalizing the denominator - Conjugates -
Simplify the radical expression. 4. 6 5 5. 8 1 6. 4 5 7. 5 3 7 Use square roots to solve the following quadratic equations. 8. 3x = 75 9. z 7 = 5 10. x 6 10 5 11. 4(x 1) 8
Mission 6 Complete the Square Review Question 1. Given the diagram below, approximate to the nearest foot how many feet of walking distance a person saves by cutting across the lawn instead of walking on the sidewalk. A. 60 feet C. 36 feet B. 48 feet D. 4 feet Objective: SWBAT complete the square. SWBAT factor by completing the square. SWBAT solve quadratics by completing the square. SWBAT derive the quadratic formula. Just Watch!!! x 0x + 4 = 0 Perfect square trinomials - Solve the equation by finding the square roots. 1. x 16x 64 36. x x 11 13
Find the c value that would make the following a perfect trinomial. 3. x 10x c 4. x 8x c 5. x 13x c Solve the following by completing the square. 6. x 1x 18 0 7. x 10x 1 0 8. x 8x 1 9. 3x 4x 4 10. x 5x 1 0 11. ax bx c 0
Mission 7 Use the Quadratic Formula and the Discriminant Review Questions 1. A manufacturer is going to package their product in an open rectangular box made from a single flat piece of cardboard. The box will be created by cutting a square out from each corner of the rectangle and folding the flaps up to create a box. The original rectangular piece of cardboard is 0 inches long and 15 inches wide. Write a function that represents the volume of the box. A. V(x) = x 3 35x + 300x C. B. V(x) = 4x 3 70x + 300x D. V(x) = x 35x + 300 V(x) = 4x 70x + 300. Simplify (8 7i) (6 + 9i) 3. Simplify i 33 + i Objective: SWBAT find and use the discriminant. SWBAT use the quadratic formula. Quadratic Formula - Discriminant - Value of the discriminant Zero Number and type of solutions Graph Positive discriminant Negative discriminant
Find the discriminant and describe the solutions. 1. 3x 11x 4 0. 3x 11x 4 0 Use the quadratic formula to solve the following equations. Give exact answers, no decimals. 3. x 1x 3 0 4. x 4x 1 0 5. 3x 11x 4 0 6. a 13a 7 7. x x 1 0 8. x 1x 1 4x 3