Equations and Roots: Complex or Real?

2. 1. Which of the following equations have complex roots? Justify your answer. (i) No real roots. Because it doesn’t cross the x-axis. Therefore, it has complex roots.

3. 1. Which of the following equations have complex roots? Justify your answer. (ii) Real roots. Roots = 2, 6

4. 1. Which of the following equations have complex roots? Justify your answer. (iii) No real roots. Because it doesn’t cross the x-axis. Therefore, it has complex roots.

5. 1. Which of the following equations have complex roots? Justify your answer. (iv) Two equal real roots. Root = 1

6. 1. Which of the following equations have complex roots? Justify your answer. (v) No real roots. Because it doesn’t cross the x-axis. Therefore, it has complex roots.

7. 1. Which of the following equations have complex roots? Justify your answer. (vi) No real roots. Because it doesn’t cross the x-axis. Therefore, it has complex roots.

8. 2. Find the roots of the following equations: (i) z2 = − 64 z2 = − 64 z2 + 64 = 0 a = 1 b = 0 c = 64 Alternatively, z2 = − 64

9. 2. Find the roots of the following equations: (ii) z2 = − 48 z2 = − 48 z2 + 48 = 0 a = 1 b = 0 c = 48 Alternatively,

10. 2. Find the roots of the following equations: (iii) z2 = − 28 z2 = − 28 z2 + 28 = 0 a = 1 b = 0 c = 28 Alternatively,

11. 2. Find the roots of the following equations: (iv) 2z2 + 50 = 0 a = 2 b = 0 c = 50 Alternatively, 2z2 + 50 = 0 2z2 = −50 z2 = −25

12. 2. Find the roots of the following equations: (v) 3z2 + 147 = 0 a = 3 b= 0 c = 147 Alternatively, 3z2 + 147 = 0 3z2 = −147 z2 = −49

13. 2. Find the roots of the following equations: (vi) 5z2 + 495 = 0 a = 5 b = 0 c = 495 Alternatively, 5z2 + 495 = 0 5z2 = −495 z2 = −99

14. 3. Find the roots of the following equations: (i) z2− 10z + 34 = 0 a = 1 b = − 10 c = 34

15. 3. Find the roots of the following equations: (ii) z2− 10z + 28 = 0 a = 1 b = − 10 c = 28

16. 3. Find the roots of the following equations: (iii) z2− 4z + 13 = 0 a = 1 b = − 4 c = 13

17. 3. Find the roots of the following equations: (iv) z2 + 5 = 4z z2− 4z + 5 = 0 a = 1 b = −4 c = 5

18. 3. Find the roots of the following equations: (v) z2− 8z = − 40 z2− 8z + 40 = 0 a = 1 b = − 8 c = 40

19. 3. Find the roots of the following equations: (vi) z2 = − 10z− 29 z2 + 10z + 29 = 0 a = 1 b = 10 c = 29

20. 4. Find the roots of the following equations: (i) 2z2 − 4z + 5 = 0 a = 2 b = − 4 c = 5

21. 4. Find the roots of the following equations: (ii) 5z2 + 12z + 8 = 0 a = 5 b = 12c = 8

22. 4. Find the roots of the following equations: (iii) 4z2 – 6z + 5 = 0 a = 4 b = − 6 c = 5

23. 4. Find the roots of the following equations: (iv) 5z2 – 6z + 9 = 0 a = 5 b = − 6 c = 9

24. 4. Find the roots of the following equations: (v) 9z2 – 6z + 5 = 0 a = 9 b = − 6 c = 5

25. 4. Find the roots of the following equations: (vi) 8z2 – 12z + 5 = 0 a = 8 b = − 12 c = 5

26. 5. Solve the equation z2 + 6z + 20 = 0 Express the solutions in the form , where a and bare integers. a = 1 b = 6 c = 20

27. 6. Use this information to determine the nature of the roots of the following equations: The quadratic formula for finding the roots of an equation is: The part of the quadratic formula under the square root sign (b2 − 4ac) is called the discriminant. The value of the discriminant can be used to determine the nature of the roots of an equation as follows: b2 − 4ac < 0 means complex roots; b2 − 4ac > 0 means two distinct real roots; b2 − 4ac = 0 means two equal real roots. (i) 6a2 – 2a – 3 = 0 a = 6 b = − 2 c = − 3 b2 – 4ac = (− 2)2 – 4(6)(− 3) = 4 – 4(− 18) = 4 + 72 = 76 > 0 Therefore, this equation has two distinct real roots.

28. 6. Use this information to determine the nature of the roots of the following equations: The quadratic formula for finding the roots of an equation is: The part of the quadratic formula under the square root sign (b2 − 4ac) is called the discriminant. The value of the discriminant can be used to determine the nature of the roots of an equation as follows: b2 − 4ac < 0 means complex roots; b2 − 4ac > 0 means two distinct real roots; b2 − 4ac = 0 means two equal real roots. (ii) 5b2 + b – 2 = 0 a = 5 b = 1 c = −2 b2 – 4ac = (1)2 – 4(5)(−2) = 1 – 4(−10) = 1 + 40 = 41 > 0 Therefore, this equation has two distinct real roots.

29. 6. Use this information to determine the nature of the roots of the following equations: The quadratic formula for finding the roots of an equation is: The part of the quadratic formula under the square root sign (b2 − 4ac) is called the discriminant. The value of the discriminant can be used to determine the nature of the roots of an equation as follows: b2 − 4ac < 0 means complex roots; b2 − 4ac > 0 means two distinct real roots; b2 − 4ac = 0 means two equal real roots. (iii) c2 + 5c + 2 = 0 a = 1 b = 5 c = 2 b2 – 4ac = (5)2 – 4(1)(2) = 25 – 4(2) = 25 – 8 = 17 > 0 Therefore, this equation has two distinct real roots.

30. 6. Use this information to determine the nature of the roots of the following equations: The quadratic formula for finding the roots of an equation is: The part of the quadratic formula under the square root sign (b2 − 4ac) is called the discriminant. The value of the discriminant can be used to determine the nature of the roots of an equation as follows: b2 − 4ac < 0 means complex roots; b2 − 4ac > 0 means two distinct real roots; b2 − 4ac = 0 means two equal real roots. (iv) 9d2 – 3d + 2 = 0 a = 9 b = −3 c = 2 b2 – 4ac = (−3)2 – 4(9)(2) = 9 – 4(18) = 9 – 72 = −63 < 0 Therefore, this equation has complex roots.

31. 6. Use this information to determine the nature of the roots of the following equations: The quadratic formula for finding the roots of an equation is: The part of the quadratic formula under the square root sign (b2 − 4ac) is called the discriminant. The value of the discriminant can be used to determine the nature of the roots of an equation as follows: b2 − 4ac < 0 means complex roots; b2 − 4ac > 0 means two distinct real roots; b2 − 4ac = 0 means two equal real roots. (v) 4e2 – 8e + 4 = 0 a = 4 b = −8 c = 4 b2 – 4ac = (−8)2 – 4(4)(4) = 64 – 4(16) = 64 – 64 = 0 Therefore, this equation has two equal real roots.

32. 6. Use this information to determine the nature of the roots of the following equations: The quadratic formula for finding the roots of an equation is: The part of the quadratic formula under the square root sign (b2 − 4ac) is called the discriminant. The value of the discriminant can be used to determine the nature of the roots of an equation as follows: b2 − 4ac < 0 means complex roots; b2 − 4ac > 0 means two distinct real roots; b2 − 4ac = 0 means two equal real roots. (vi) 2f 2 – 10f – 5 = 0 a = 2 b = −10 c = −5 b2 – 4ac = (−10)2 – 4(2)(−5) = 100 – 4(−10) = 100 + 40 = 140 > 0 Therefore, this equation has two distinct real roots.

33. 6. Explain why b2 − 4ac < 0 results in complex roots. The quadratic formula for finding the roots of an equation is: The part of the quadratic formula under the square root sign (b2 − 4ac) is called the discriminant. The value of the discriminant can be used to determine the nature of the roots of an equation as follows: b2 − 4ac < 0 means complex roots; b2 − 4ac > 0 means two distinct real roots; b2 − 4ac = 0 means two equal real roots. b2 – 4ac < 0 results in complex roots because the number under the square root is negative. The square root of a negative number is imaginary.

34. 6. Explain why b2 − 4ac = 0 results in equal roots. The quadratic formula for finding the roots of an equation is: The part of the quadratic formula under the square root sign (b2 − 4ac) is called the discriminant. The value of the discriminant can be used to determine the nature of the roots of an equation as follows: b2 − 4ac < 0 means complex roots; b2 − 4ac > 0 means two distinct real roots; b2 − 4ac = 0 means two equal real roots. b2 – 4ac = 0 results in equal roots because and so the  is eliminated and so the two roots are equal.

35. 7. Show that 2 − 2i is a root of the equation z2 − 4z + 8 = 0. Hence, or otherwise, find the other root. z2 – 4z + 8 = 0 a = 1 b = −4 c = 8 2 – 2i is a root. The other root is 2 + 2i.

36. 8. f(x) = 8 − 4x and g(x) = x2 − 5x − 2 are two functions. (i) Solve f(x) = g(x). f(x) = 8 – 4xg(x) = x2 – 5x – 2 f(x) = g(x) 8 – 4x = x2 – 5x – 2 x2 – 5x + 4x – 2 – 8 = 0 x2 – x – 10 = 0 a = 1 b = −1 c = −10

37. 8. f(x) = 8 − 4x and g(x) = x2 − 5x − 2 are two functions. (i) Solve f(x) = g(x). = Roots are real

38. 8. f(x) = 8 − 4x and g(x) = x2 − 5x − 2 are two functions. (ii) What does your answer to part (i) tell you about the graphs of the functions? Justify your answer. The graphs intersect because f(x) = g(x) has real roots.

39. 9. Verify that the function f(x) = x2 + 2x + 5 does not cross the -axis (i.e. has complex roots). f(x) = x2 + 2x + 5 b2 – 4ac < 0 if the roots are complex a = 1 b = 2 c = 5 b2 – 4ac = 22 – 4(1)(5) = 4 – 20 = − 16 < 0 Therefore, this function has complex roots.

40. 9. Verify that the function f(x) = x2 + 2x + 5 does not cross the -axis (i.e. has complex roots). Alternatively, Roots = −1  2i Function has complex roots and so does not cross the x-axis.

41. 10. Investigate whether the equation 2x2 − 3x − 1 = 0 has real or complex roots. Justify your answer. 2x2 – 3x – 1 = 0 a = 2 b = −3 c = −1 We examine b2 – 4ac. b2 – 4ac = (−3)2 – 4(2)(−1) = 9 – 4(−2) = 9 + 8 = 17 > 0 Therefore, this equation has two distinct real roots.

42. 10. Investigate whether the equation 2x2 − 3x − 1 = 0 has real or complex roots. Justify your answer. 2x2 – 3x – 1 = 0 a = 2 b = −3 c = −1 Alternatively, Therefore, this equation has 2 real roots.