1. Martin-Gay, Developmental Mathematics 1 Warm Up Factor the following

2. Solving Quadratic Equations by the Square Root Property

3. Martin-Gay, Developmental Mathematics 3 Square Root Property We previously have used factoring to solve quadratic equations. This chapter will introduce additional methods for solving quadratic equations. Square Root Property If b is a real number and a 2 = b, then

4. Martin-Gay, Developmental Mathematics 4 Solve x 2 = 49 Solve (y – 3) 2 = 4 Solve 2x 2 = 4 x 2 = 2 y = 3  2 y = 1 or 5 Square Root Property Example

5. Martin-Gay, Developmental Mathematics 5 Solve x 2 + 4 = 0 x 2 =  4 There is no real solution because the square root of  4 is not a real number. Square Root Property Example

6. Martin-Gay, Developmental Mathematics 6 Solve (x + 2) 2 = 25 x =  2 ± 5 x =  2 + 5 or x =  2 – 5 x = 3 or x =  7 Square Root Property Example

7. Martin-Gay, Developmental Mathematics 7 Solve (3x – 17) 2 = 28 3x – 17 = Square Root Property Example

8. Solving Quadratic Equations by the Quadratic Formula https://www.youtube.com/watch?v=YC uXiujC3KE

9. Martin-Gay, Developmental Mathematics 9 The Quadratic Formula Another technique for solving quadratic equations is to use the quadratic formula. The formula is derived from completing the square of a general quadratic equation.

10. Martin-Gay, Developmental Mathematics 10 Quadratic Formula If we are unable to factor a quadratic function to find the roots we can utilize the Quadratic Formula. The entire equation can tell us the number of roots & the radicand tells us the number of real solutions. A quadratic equation written in standard form, ax 2 + bx + c = 0, has the solutions

11. Martin-Gay, Developmental Mathematics 11 Discriminant Positive, perfect squares2 real, rational roots Positive, not perfect squares2 real, irrational roots Zero1 real rational root Negative2 complex roots

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16. Martin-Gay, Developmental Mathematics 16 Solution

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19. Martin-Gay, Developmental Mathematics 19 Solve 11n 2 – 9n = 1 by the quadratic formula. 11n 2 – 9n – 1 = 0, so a = 11, b = -9, c = -1 The Quadratic Formula Example

20. Martin-Gay, Developmental Mathematics 20 x 2 + 8x – 20 = 0 (multiply both sides by 8) a = 1, b = 8, c =  20 Solve x 2 + x – = 0 by the quadratic formula. The Quadratic Formula Example

21. Martin-Gay, Developmental Mathematics 21 Solve x(x + 6) =  30 by the quadratic formula. x 2 + 6x + 30 = 0 a = 1, b = 6, c = 30 So there is no real solution. The Quadratic Formula Example

22. Martin-Gay, Developmental Mathematics 22 The expression under the radical sign in the formula (b 2 – 4ac) is called the discriminant. The discriminant will take on a value that is positive, 0, or negative. The value of the discriminant indicates two distinct real solutions, one real solution, or no real solutions, respectively. The Discriminant

23. Martin-Gay, Developmental Mathematics 23 Use the discriminant to determine the number and type of solutions for the following equation. 5 – 4x + 12x 2 = 0 a = 12, b = –4, and c = 5 b 2 – 4ac = (–4) 2 – 4(12)(5) = 16 – 240 = –224 There are no real solutions. The Discriminant Example

24. Martin-Gay, Developmental Mathematics 24 Solving Quadratic Equations Steps in Solving Quadratic Equations 1)If the equation is in the form (ax+b) 2 = c, use the square root property to solve. 2)If not solved in step 1, write the equation in standard form. 3)Try to solve by factoring. 4)If you haven’t solved it yet, use the quadratic formula.

25. Martin-Gay, Developmental Mathematics 25 Solve 12x = 4x 2 + 4. 0 = 4x 2 – 12x + 4 0 = 4(x 2 – 3x + 1) Let a = 1, b = -3, c = 1 Solving Equations Example

26. Martin-Gay, Developmental Mathematics 26 Solve the following quadratic equation. Solving Equations Example

27. § 16.4 Graphing Quadratic Equations in Two Variables

28. Martin-Gay, Developmental Mathematics 28 We spent a lot of time graphing linear equations in chapter 3. The graph of a quadratic equation is a parabola. The highest point or lowest point on the parabola is the vertex. Axis of symmetry is the line that runs through the vertex and through the middle of the parabola. Graphs of Quadratic Equations

29. Martin-Gay, Developmental Mathematics 29 x y Graph y = 2x 2 – 4. x y 0 –4–4 1 –2–2 –1–1 –2–2 24 –2–24 (2, 4) (–2, 4) (1, –2)(–1, – 2) (0, –4) Graphs of Quadratic Equations Example

30. Martin-Gay, Developmental Mathematics 30 Although we can simply plot points, it is helpful to know some information about the parabola we will be graphing prior to finding individual points. To find x-intercepts of the parabola, let y = 0 and solve for x. To find y-intercepts of the parabola, let x = 0 and solve for y. Intercepts of the Parabola

31. Martin-Gay, Developmental Mathematics 31 If the quadratic equation is written in standard form, y = ax 2 + bx + c, 1) the parabola opens up when a > 0 and opens down when a < 0. 2) the x-coordinate of the vertex is. To find the corresponding y-coordinate, you substitute the x-coordinate into the equation and evaluate for y. Characteristics of the Parabola

32. Martin-Gay, Developmental Mathematics 32 x y Graph y = – 2x 2 + 4x + 5. x y 1 7 2 5 05 3–1 (3, –1) (–1, –1) (2, 5) (0, 5) (1, 7) Since a = –2 and b = 4, the graph opens down and the x-coordinate of the vertex is Graphs of Quadratic Equations Example

33. § 16.5 Interval Notation, Finding Domain and Ranges from Graphs, and Graphing Piecewise-Defined Functions

34. Martin-Gay, Developmental Mathematics 34 Recall that a set of ordered pairs is also called a relation. The domain is the set of x-coordinates of the ordered pairs. The range is the set of y-coordinates of the ordered pairs. Domain and Range

35. Martin-Gay, Developmental Mathematics 35 Find the domain and range of the relation {(4,9), (–4,9), (2,3), (10, –5)} Domain is the set of all x-values, {4, –4, 2, 10} Range is the set of all y-values, {9, 3, –5} Example Domain and Range

36. Martin-Gay, Developmental Mathematics 36 Find the domain and range of the function graphed to the right. Use interval notation. x y Domain is [ – 3, 4] Domain Range is [ – 4, 2] Range Example Domain and Range

37. Martin-Gay, Developmental Mathematics 37 Find the domain and range of the function graphed to the right. Use interval notation. x y Domain is (– ,  ) Domain Range is [– 2,  ) Range Example Domain and Range

38. Martin-Gay, Developmental Mathematics 38 Input (Animal) Polar Bear Cow Chimpanzee Giraffe Gorilla Kangaroo Red Fox Output (Life Span) 20 15 10 7 Find the domain and range of the following relation. Example Domain and Range

39. Martin-Gay, Developmental Mathematics 39 Domain is {Polar Bear, Cow, Chimpanzee, Giraffe, Gorilla, Kangaroo, Red Fox} Range is {20, 15, 10, 7} Domain and Range Example continued

40. Martin-Gay, Developmental Mathematics 40 Graph each “piece” separately. Graph Graphing Piecewise-Defined Functions Example Continued. x f (x) = 3x – 1 0– 1 (closed circle) –1– 4 –2– 7 x f (x) = x + 3 1 4 2 5 3 6 Values  0. Values > 0.

41. Martin-Gay, Developmental Mathematics 41 Example continued Graphing Piecewise-Defined Functions x y x f (x) = x + 3 1 4 2 5 3 6 x f (x) = 3x – 1 0– 1 (closed circle) –1– 4 –2– 7 (0, –1) (–1, 4) (–2, 7) Open circle (0, 3) (3, 6)