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Solving Quadratic Functions

Solving Quadratic Functions. Lesson 5-3. Objective. Today, you will . . . solve quadratic functions by using a variety of methods. TEKS:b2A,d1A,d3A,d3C,d3D. Some Notes on Quadratic Functions. The graphs of quadratic functions are parabolas.

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Solving Quadratic Functions

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  1. Solving Quadratic Functions Lesson 5-3

  2. Objective • Today, you will . . . • solve quadratic functions by using a variety of methods. TEKS:b2A,d1A,d3A,d3C,d3D

  3. Some Notes on Quadratic Functions • The graphs of quadratic functions are parabolas. • The solution(s) that you will be looking for are the x-intercepts of the parabola. • The x-intercepts are also called the “roots,” “solutions,” or the “zeros” • Quadratic functions can have one, two, or no real solutions. • Quadratic functions that have no real solutions have complex (imaginary) solutions.

  4. Different Graphs of Quadratics Two real solutions X=-6 X=1 x-intercepts, roots, zeros

  5. Different Graphs of Quadratics One real solutions X=3 x-intercept, root, zero

  6. Different Graphs of Quadratics No real solutions No Solution No x-intercepts (roots or zeros)

  7. Today you’ll find… • The solutions to quadratic equations by factoring • For example: GIVEN: y = Ax2 + Bx + C FIND:The solutions, roots, zeros, or x-intercepts

  8. Solve by factoring: x2 + 9x + 20 = 0 1 x 20 2 x 20 4 x 5 x2 + 9x + 20 = 0 (x + 5)(x + 4) = 0 x + 5 = 0 x + 4 = 0 x = - 5 x = - 4 So, its two roots, solutions, zeros are -5 & -4

  9. Solve by factoring: x2 = -7x + 18 x2 + 7x - 18 = 0 1 x 18 2 x 9 3 x 6 Hint: The sign of “B” goes with the largest factor! (x - 2)(x + 9) = 0 x + 9 = 0 x - 2 = 0 x = 2 x = - 9 So, its two roots, solutions, zeros are 2 & -9

  10. Solve by factoring: x2 + 100 = 29x 1 x 100 2 x 50 4 x 25 5 x 20 10 x 10 x2 - 29x + 100 = 0 (x - 4)(x - 25) = 0 x - 4 = 0 x - 25 = 0 x = 4 x = 25 So, its two roots, solutions, zeros are 4& 25

  11. Solve by factoring: x2 - 9 = 0 1 x 9 3 x 3 x2 - 9 = 0 (x + 3)(x - 3) = 0 x + 3 = 0 x - 3 = 0 x = - 3 x = 3 So, its two roots, solutions, zeros are -3 & 3

  12. Solve by factoring: x2 + x = 6 1 x 6 2 x 3 x2 + x - 6 = 0 Hint: The sign of “B” goes with the largest factor! (x - 2)(x + 3) = 0 x - 2 = 0 x + 3 = 0 x = 2 x = - 3 So, its two roots, solutions, zeros are 2 & -3

  13. Solve by factoring: x2 - 6x = - 8 x = 2 1 x 8 2 x 4 x = 4 Solve by factoring: x2 - 8 = - 7x x = 1 1 x 8 2 x 4 x = - 8

  14. Your solutions are x = -3/2 and x = -5 Solve byfactoring: 2x2 + 13x + 15 = 0 2x2 + 13x + 15 = 0 Multiply AxC = 30 1 x 30 2 x 15 3 x 10 Determine the factors of 30 that give you 13 (x + 3) (x + 10) Write the factors (x + 3) (x + 10 ) Divide the #’s by A 2 2 (2x + 3) (x + 5) If not divisible, send it in front of “x”, if divisible then simplify. 2x + 3 = 0 x + 5 = 0Now solve both factors! 2x = -3 x = -5 x = -3/2

  15. Solve byfactoring: 3x2 + 16x + 21 = 0 3x2 + 16x + 21 = 0 Multiply AxC = 63 1 x 63 3 x 21 7 x 9 Determine the factors of 63 that give you 16 (x + 7) (x + 9) Write the factors (x + 7) (x + 9 ) Divide the #’s by A 3 3 (3x + 7) (x + 3) If not divisible, send it in front of “x”, if divisible then simplify. 3x + 7 = 0 x + 3 = 0Now solve both factors! 3x = -7 x = -3 x = -7/3 Your solutions are x = -7/3 and x = -3

  16. Solve byfactoring: 2x2 – 5x = 7 2x2 – 5x – 7 = 0 Multiply AxC = -14 1 x 14 2 x 7 Determine the factors of -14 that give you -5 (x + 2) (x – 7) Write the factors Remember, sign of “B” goes to the largest factor, in this case, the negative goes to the 7. (x + 2) (x – 7) Divide the #’s by A 2 2 (x + 1) (2x – 7) If not divisible, send it in front of “x”, if divisible then simplify. x + 1 = 0 2x – 7 = 0Now solve both factors! x = -1 2x = 7 x = 7/2 Your solutions are x = -1 and x = 7/2

  17. Your turn!Solve byfactoring: 5x2 + 4x = 12 5x2 + 4x – 12 = 0 A x C = -60 1 x 60 4 x 15 2 x 30 5 x 12 3 x 20 6 x 10 (x – 6) (x + 10) Write the factors (watch your signs!) (x – 6) (x + 10) Divide the #’s by A 5 5 (5x – 6)(x + 2) If not divisible, send it in front of “x”, if divisible then simplify. 5x – 6 = 0 x + 2 = 0Now solve both factors! 5x = 6 x = -2 x = 6/5 Your solutions are x = 6/5 and x = -2

  18. You try these by factoring . . . 1. 2x2 – 3x = 0 x = 0 and 3/2 2. 4x2 + 5 = -9x x = -1 and -5/4 3. 6x2 + 55x = -9 x = -9 and -1/6

  19. Questions ?

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