In Exercises 1–6, solve the equation Ax = b by using the LU factorization given for A. In Exercises 1 and 2, also solve Ax = b by ordinary row reduction.
6.
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- For Exercises 115–120, factor the expressions over the set of complex numbers. For assistance, consider these examples. • In Section R.3 we saw that some expressions factor over the set of integers. For example: x - 4 = (x + 2)(x – 2). • Some expressions factor over the set of irrational numbers. For example: - 5 = (x + V5)(x – V5). To factor an expression such as x + 4, we need to factor over the set of complex numbers. For example, verify that x + 4 = (x + 2i)(x – 2i). 115. а. х - 9 116. а. х? - 100 117. а. х - 64 b. x + 9 b. + 100 b. x + 64 118. а. х — 25 119. а. х— 3 120. а. х — 11 b. x + 25 b. x + 3 b. x + 11arrow_forwardFind the general solutions of the systems whose augmented matrices are given in Exercises 7–14.arrow_forwardFind a and b if a + (5 – 3b) i = ai + 2 (b + 5)arrow_forward
- In Exercises 14–16, divide as indicated. 14. (12x*y³ + 16x?y³ – 10x²y²) ÷ (4x?y) 15. (9x – 3x2 – 3x + 4) ÷ (3x + 2) 16. (3x4 + 2x3 – 8x + 6) ÷ (x² – 1)arrow_forwardFor Exercises 8–10, a. Simplify the expression. Do not rationalize the denominator. b. Find the values of x for which the expression equals zero. c. Find the values of x for which the denominator is zero. 4x(4x – 5) – 2x² (4) 8. -6x(6x + 1) – (–3x²)(6) (6x + 1)2 9. (4x – 5)? - 10. V4 – x² - -() 2)arrow_forwardIn Problems 75–84, find the real solutions of each equation.arrow_forward
- In Problems 11–48, perform the indicated operation, and write each expression in the standard form a + bi. 11. (2 — 3і) + (6 + 8i) 12. (4 + 5i) + (-8 + 2i) 13. (-3 + 2i) – (4 – 4i) 14. (3 – 4i) – (-3 – 4i) 15. (2 — 51) — (8+ 6і) 16. (-8 + 4i) – (2 – 2i) 17. 3(2 — 6і) 18. –4(2 + 8i) 19. 2i (2 — 3і) 20. Зi(-3 + 4i) \ 21. (3 – 4i) (2 + i) 22. (5 + 3i) (2 - i) 10 13 23. (-6 + i) (-6 — і) 24. (-3 + i) (3 + i) 25. 3 26. 5 - 12i 4i 2 + i 27. 2 - i 28. -2i 6 - i 29. 1 + i 2 + 3i 30. 1 - i V3 31. 33. (1 + i)? 34. (1 – i)? SECTION A.7 Complex Numbers; Quadratic Equations in the Complex Number System 35. Р3 36. i14 37. г15 38. 23 39. i6 – 5 41. 6i – 4i5 43. (1 + i) 48. i + + P + i 40. 4 + 3 42. 4i – 21 + 1 44. (3i)* + 1 45. i' (1 + ?) 46. 2i* (1 + ?) 47. * + i* + ² + 1arrow_forward2 (8a 3) 3 in simplest form.arrow_forwardSolve the ytem 2x -44 =6 X +Z +y=' ダ+ダ= /o Solarrow_forward
- Exercises 43–52: Complete the following. (a) Solve the equation symbolically. (b) Classify the equation as a contradiction, an identity, or a conditional equation. 43. 5x - 1 = 5x + 4 44. 7- 9: = 2(3 – 42) – z 45. 3(x - 1) = 5 46. 22 = -2(2x + 1.4) 47. 0.5(x – 2) + 5 = 0.5x + 4 48. 눈x-2(x-1)3-x + 2 2x + 1 2x 49. 50. x – 1.5 2- 3r - 1.5 51. -6 52. 0.5 (3x - 1) + 0.5x = 2x – 0.5arrow_forward8.arrow_forwardIn Exercises 49–55, solve each rational equation. If an equation has no solution, so state. 3 1 + 3 49. 3 50. Зх + 4 2x - 8 1 3 6. 51. x + 5 x² 25 x + 5 52. x + 1 4x + 1 x + 2 x2 + 3x + 2 2 53. 3 - 3x .2 2 7 54. 4 x + 2 2x + 7 55. x + 5 8. x + 18 x - 4 x + x - 20arrow_forward
- Algebra & Trigonometry with Analytic GeometryAlgebraISBN:9781133382119Author:SwokowskiPublisher:Cengage