Bài tập cơ bản giải phương trình vô tỷ – Toán 9

Bài tập giải phương trình vô tỷ mức độ cơ bản dành cho học sinh lớp 9.

Bài 1: Giải các phương trình sau:
1. $\displaystyle\sqrt{{{x}^{2}}-6x+9}=x$

2. $\displaystyle\sqrt{4{{x}^{2}}-12x+9}=x-1$

3. $x+\sqrt{4{{x}^{2}}-4x+1}=2$

4. $1-\sqrt{4{{x}^{4}}-20{{x}^{2}}+25}=0$

5. ${{x}^{2}}-\sqrt{{{x}^{2}}}=0$

6. ${{x}^{2}}+\sqrt{{{x}^{2}}}=0$

7. $x+\sqrt{{{x}^{2}}-4x+4}=0$

8. $x-\sqrt{4{{x}^{2}}-12x+9}=0$

9. $3x-1-\sqrt{4{{x}^{2}}-12x+9}=0$

10. $x-\sqrt{4{{x}^{2}}-12x+9}=3$

11. $\displaystyle\sqrt{3-2\sqrt{2}}-\sqrt{{{x}^{2}}+2x\sqrt{3}+3}=0$

12. $\displaystyle\sqrt{5{{x}^{2}}-2x\sqrt{5}+1}=\sqrt{6-2\sqrt{5}}$

13. $\displaystyle\sqrt{4{{x}^{2}}+4x\sqrt{7}+7}-\sqrt{8-2\sqrt{7}}=0$

14. $\displaystyle\sqrt{7-2\sqrt{10}}-\sqrt{5{{x}^{2}}-2x\sqrt{10}+2}=0$

15. $\displaystyle\sqrt{11+6\sqrt{2}}=\sqrt{2{{x}^{2}}-6x\sqrt{2}+9}$

16. $\displaystyle\sqrt{11-\sqrt{120}}=\sqrt{5{{x}^{2}}+x\sqrt{120}+6}$

17. $\displaystyle\sqrt{1+2x\sqrt{3}+3{{x}^{2}}}-\sqrt{3+2x\sqrt{3}+{{x}^{2}}}=0$

18. $\displaystyle\sqrt{5{{x}^{2}}-2x\sqrt{5}+1}-\sqrt{4{{x}^{2}}+4x\sqrt{5}+5}=0$

19. $\displaystyle\sqrt{16{{x}^{2}}+8x\sqrt{2}+2}-\sqrt{9{{x}^{2}}-6x\sqrt{2}+2}=0$

20. $\displaystyle\sqrt{2{{x}^{2}}-2x\sqrt{6}+3}-\sqrt{2-2x\sqrt{6}+3{{x}^{2}}}=0$

21. $\displaystyle\sqrt{8{{x}^{2}}-4x\sqrt{2}+1}-\sqrt{{{x}^{2}}-6x\sqrt{2}+18}=0$

22. $\displaystyle\sqrt{5{{x}^{2}}+2x\sqrt{30}+6}-\sqrt{6{{x}^{2}}+2x\sqrt{30}+5}=0$

23. $\displaystyle\sqrt{{{x}^{2}}}=x$

24. $\displaystyle\sqrt{{{x}^{2}}-2x+1}=x-1$

25. $\displaystyle\sqrt{{{x}^{2}}-4x+4}=x-2$

26. $\displaystyle\sqrt{16-8x+{{x}^{2}}}=4-x$

27. $\displaystyle\sqrt{4{{x}^{2}}-12x+9}=2x-3$

28. $\displaystyle\sqrt{25{{x}^{2}}-10x+x}=5x-1$

29. $\displaystyle\sqrt{{{x}^{2}}-2x\sqrt{5}+5}=x-\sqrt{5}$

30. $\displaystyle\sqrt{3{{x}^{2}}-6x\sqrt{2}+6}=\sqrt{3}x-\sqrt{6}$

31. $\displaystyle\sqrt{10{{x}^{2}}-12x\sqrt{10}+36}=\sqrt{10}x-6$

32. $\displaystyle\sqrt{7{{x}^{2}}+2x\sqrt{14}+2}=\sqrt{7}x+\sqrt{2}$

33. $\displaystyle\sqrt{{{x}^{2}}}=-x$

34. $\displaystyle\sqrt{{{x}^{2}}-6x+9}=3-x$

35. $\displaystyle\sqrt{{{x}^{2}}-4x+4}=2-x$

36. $\displaystyle\sqrt{{{x}^{2}}+4x+4}=-x-2$

37. $\displaystyle\sqrt{4{{x}^{2}}+4x+1}=-2x-1$

38. $\displaystyle\sqrt{{{x}^{2}}+x+\frac{1}{4}}=-x-\frac{1}{2}$

39. $\displaystyle\sqrt{x+2\sqrt{x}+1}-\sqrt{x-2\sqrt{x}+1}=2$

40. $\displaystyle\sqrt{x+4\sqrt{x}+4}+\sqrt{x-4\sqrt{x}+4}=4$

41. $\displaystyle\sqrt{x+6\sqrt{x}+9}-6=\sqrt{9-6\sqrt{x}+x}$

42. $\displaystyle\sqrt{4x+4\sqrt{x}+1}=\sqrt{1-4\sqrt{x}+4x}+2$

43. $\displaystyle\sqrt{x-2\sqrt{x-1}}-\sqrt{x+2\sqrt{x-1}}=-2$

44. $\displaystyle\sqrt{x-2\sqrt{x-2}-1}-\sqrt{x+2+4\sqrt{x-2}+3}=0$

45. $-\sqrt{x+3+4\sqrt{x-1}}+\sqrt{x+8-6\sqrt{x-1}}=-5$

46. $\displaystyle\sqrt{x+4\sqrt{x-4}}-\sqrt{x-4\sqrt{x-4}}=4$

47. $-\sqrt{2x+4+6\sqrt{2x-5}}+\sqrt{2x-4-2\sqrt{2x-5}}=-4$

48. $4+\sqrt{2x+6-6\sqrt{2x-3}}=\sqrt{2x-2+2\sqrt{2x-3}}$

49. $\displaystyle\sqrt{x+2\sqrt{x}+1}+\sqrt{x-2\sqrt{x}+1}=2$

50. $\displaystyle\sqrt{x-2\sqrt{x-1}}+\sqrt{x+2\sqrt{x-1}}=2$

51. $\displaystyle\sqrt{x-2\sqrt{x-2}-1}+\sqrt{x+2+4\sqrt{x-2}}-3=0$

52. $\displaystyle\sqrt{x+3+4\sqrt{x-1}}+\sqrt{x+8-6\sqrt{x-1}}=5$

53. $\displaystyle\sqrt{x+4\sqrt{x-4}}+\sqrt{x-4\sqrt{x-4}}=4$

54. $\displaystyle\sqrt{2x+4+6\sqrt{2x-5}}+\sqrt{2x-4-2\sqrt{2x-5}}=4$

55. $\displaystyle\sqrt{2x+2\sqrt{2x}+1}-\sqrt{2x-2\sqrt{2x}+1}=2$

56. $\displaystyle\sqrt{4+4\sqrt{3x}+3x}-4=\sqrt{4-4\sqrt{3x}+3x}$

57. $\displaystyle\sqrt{12x-4\sqrt{3}x+1}-\sqrt{1+4\sqrt{3}x+12x}=-2$

58. $\displaystyle\sqrt{18x+6\sqrt{2}x+1}-\sqrt{1-6\sqrt{2}x+18x}=2$ .

Bài 2: Giải các phương trình sau:

1. $\displaystyle\sqrt{2x-3}=\sqrt{x-1}$

2. $\displaystyle\sqrt{2x-3}-\sqrt{x+3}=0$

3. $\displaystyle\sqrt{x-1}=\sqrt{2x+3}$

4. $\displaystyle\sqrt{2x-3}=\sqrt{x+1}$

5. $\displaystyle\sqrt{x+2}=\sqrt{2x-4}$

6. $\displaystyle\sqrt{2-x}=\sqrt{3+x}$

7. $\displaystyle\sqrt{1-x}-\sqrt{x-3}=0$

8. $\displaystyle\sqrt{-2x}-\sqrt{2-x}=0$

9. $\displaystyle\sqrt{6-x}-\sqrt{-3x}=0$

10. $\displaystyle\sqrt{{{x}^{2}}-3x}-\sqrt{5\left( 3-x \right)}=0$

11. $\displaystyle\sqrt{x-2}=\sqrt{x-2}$

12. $\displaystyle\ \sqrt{4x-8}=2\sqrt{x-2}$

13. $\displaystyle\sqrt{x-1}-\sqrt{x-4}=0$

14. $\displaystyle\sqrt{2-x}-\sqrt{3-x}=0$

15. $\displaystyle\sqrt{{{x}^{2}}-5}=\sqrt{4x-9}$

16. $\displaystyle\sqrt{2{{x}^{2}}-6x+2}=\sqrt{{{x}^{2}}-3x}$

17. $\displaystyle\sqrt{{{x}^{2}}-2x-4}=\sqrt{2-x}$

18. $\displaystyle\sqrt{{{x}^{2}}-x-1}=\sqrt{x-1}$

19. $\displaystyle\sqrt{x-2}-\sqrt{{{x}^{2}}-2x}=0$

20. $\displaystyle\sqrt{{{x}^{2}}-x-2}=\sqrt{x+1}$

21. $\displaystyle\sqrt{2{{x}^{2}}-10x+11}=\sqrt{{{x}^{2}}-6x+8}$

22. $\displaystyle\sqrt{2{{x}^{2}}+6x-3}=\sqrt{{{x}^{2}}+4x}$

23. $\displaystyle\sqrt{2{{x}^{2}}+x-9}=\sqrt{{{x}^{2}}-x-6}$

24. $\displaystyle\sqrt{x-1}=2$

25. $\displaystyle\sqrt{2x-3}=13$

26. $\displaystyle\sqrt{2x-3}=\sqrt{2}$

27. $\displaystyle\sqrt{x\left( x-2 \right)}-\sqrt{3}=0$

28. $3\sqrt{x}=-1$

29. $\displaystyle\sqrt{x-1}+2=0$

30. $6-\sqrt{2x+3}=12$

31. $3\sqrt{{{x}^{2}}-x}-\sqrt{54}=0$

32. $2\sqrt{2}-\sqrt{{{x}^{2}}+2x}=0$

33. $2\sqrt{3}-\sqrt{7x-{{x}^{2}}}=0$

34. $3-\sqrt{{{x}^{2}}+3}=0$

35. $2-\sqrt{x\left( 4-x \right)}=0$

36. $3-\sqrt{-x\left( x+6 \right)}=0$

37. $2-\sqrt{{{x}^{2}}-1}=0$

38. $1-\sqrt{{{x}^{2}}-2}=0$

39. $\displaystyle\sqrt{16}-2\sqrt{{{x}^{2}}+3x}=0$

40. $2\sqrt{3}-\sqrt{x\left( x+7 \right)}=0$

41. $\displaystyle\sqrt{3-x}=3x-5$ ( PTNK, CD, 1999-2000)

42. $x-\sqrt{4x-3}=2$ ( PTNK, AB, 2004-2005, Vòng 1)

43. $\displaystyle\sqrt{{{x}^{2}}-x}=x$

44. $\displaystyle\sqrt{{{x}^{2}}-1}=x-1$

45. $\displaystyle\sqrt{3-{{x}^{2}}}=x$

46. $\displaystyle\sqrt{{{x}^{2}}-2x+2}=x-1$

47. $\displaystyle\sqrt{5-{{x}^{2}}}=x-1$ ( LÊ HỒNG PHONG, 2006-2007, Vòng 1)

48. $x-2=\sqrt{{{x}^{2}}-4x+3}$

49. $\displaystyle\sqrt{x}-x=0$

50. $\displaystyle\sqrt{x}+x=0$

51. $x-\sqrt{2x-9}=6$

52. $2x-\sqrt{4x-1}=0$

53. $3x-\sqrt{6x-1}=0$

54. $x-2\sqrt{x-1}=16$

55. $x+\sqrt{-\left( 2x+1 \right)}=0$

56. $x+2\sqrt{x-1}=0$

57. $x+\sqrt{3\left( 6-x \right)}=0$

58. $x+\sqrt{x+3}=0$

59. $x+\sqrt{2x+3}=0$

60. $x+\sqrt{5-4x}=0$

61. $2x+\sqrt{3x+7}=0$

62. $3x+\sqrt{5x+4}=0$

63. $2x-\sqrt{x\left( 1-2x \right)}=1$

64. $x+\sqrt{1-{{x}^{2}}}=1$

65. $x+\sqrt{4-{{x}^{2}}}=2$

66. $2x+\sqrt{4\left| x \right|-1}=0$

67. $x+2\sqrt{\left| x \right|-1}=0$

68. $-5x+\sqrt{2\left| x \right|+3}=0$

69. $3\sqrt{-2\left| x \right|+1}=9x$

70. $7x+\sqrt{\left| x \right|-4}=0$

71. $\displaystyle\sqrt{10\left| x \right|-10}-6x=0$

72. $\displaystyle\sqrt{-3\left| x \right|+2}+1=x$

73. $\displaystyle\sqrt{7\left| x \right|+11}+x+1=0$

74. $\displaystyle\sqrt{2\left| x-1 \right|-3}-x+1=0$

75. $-3\sqrt{2\left| 2x+1 \right|-5}+6x+3=0$

76. $5\sqrt{2\left| 1-5x \right|+3}-5+25x=0$

77. $-2\sqrt{-8\left| 2-3x \right|+9}+4-6x=0$

78. $7\sqrt{-4\left| 3x-9 \right|+5}+21x-63=0$

79. $\displaystyle\sqrt{-2\left| 1-2x \right|+3}+1=2x$

80. $\displaystyle\ 3\sqrt{2x}-5\sqrt{8x}+7\sqrt{18x}=8$

81. $\displaystyle\frac{5}{3}\sqrt{15x}-\sqrt{15x}+11=\frac{1}{3}\sqrt{15x}$

82. $\displaystyle\sqrt{4x+20}-\sqrt{x+5}-\frac{1}{3}\sqrt{9x+45}=4$

83. $\displaystyle\sqrt{36x-36}-\sqrt{9x-9}-\sqrt{4x-4}=16-\sqrt{x-1}$

84. $\displaystyle\sqrt{36x-72}-\sqrt{9x-18}+\sqrt{4x-8}+\sqrt{x-2}=\sqrt{72}$

85. $\displaystyle\sqrt{9x+18}-\sqrt{x+2}-\sqrt{4x+8}+2\sqrt{x+5}=0$

86. $\displaystyle\frac{5}{3}\sqrt{45x}-\sqrt{125x}-\frac{1}{3}\sqrt{405x}-2\sqrt{16-16x}=0$

87. $\displaystyle\frac{1}{5}\sqrt{25x-125}-\frac{3}{2}\sqrt{x-5}+\sqrt{36x-180}+\sqrt{9x-27}=0$

88. $\displaystyle\sqrt{36x-216}-\sqrt{x-6}+\frac{7}{2}\sqrt{4x-24}=\frac{3}{7}\sqrt{49x-343}$

89. $15\sqrt{x-7}-2\sqrt{9x-63}-9\sqrt{25x-175}=\sqrt{4x-24}$

90. $\displaystyle\ \sqrt{49x-98}-\sqrt{9x-18}-\sqrt{16x-32}=\sqrt{4x-4}$

91. $7\sqrt{x}+\sqrt{81x-81}+\sqrt{x-1}=\sqrt{100x-100}$

92. $\displaystyle\sqrt{{{x}^{2}}-2x\sqrt{5}+5}+\sqrt{{{x}^{2}}-x\sqrt{5}}=0$

93. $\displaystyle\sqrt{0,2{{x}^{2}}-2x+5}+\sqrt{3{{x}^{2}}-15x}=0$

94. $\displaystyle\sqrt{{{x}^{2}}+4x}+\sqrt{\frac{{{x}^{2}}}{2}-8}=0$

95. $\displaystyle\sqrt{x-1}+\sqrt{{{x}^{2}}-3x+2}=0$

96. $\displaystyle\sqrt{2+x}+\sqrt{4{{x}^{2}}-6x-10}=0$

97. $\displaystyle\sqrt{{{x}^{2}}-9}+\sqrt{{{x}^{2}}-4x+3}=0$

98. $\displaystyle\sqrt{-2{{x}^{2}}+3x+5}+\sqrt{2{{x}^{2}}-7x-15}=0$

LUYỆN TẬP
Bài 3: Giải các phương trình sau:

1. ${{x}^{2}}+4\left( \left| x-2 \right|-x \right)-1=0$
a) Đặt $t=\left| x-2 \right|$ để đưa phương trình trên về phương trình theo ẩn t.
b) Tìm t rồi sau đó tìm x.

2. ${{x}^{2}}+2\left( \left| x-1 \right|-x \right)-2=0$

3. ${{x}^{2}}+3\left( \left| x-3 \right|-2x \right)-9=0$

4. ${{x}^{2}}+4\left( \left| x-4 \right|-2x \right)+4=0$

5. ${{x}^{2}}+\left( x+\left| x+1 \right| \right)-14=0$ (PTNK BAN CD 2006-2007)

6. $7-4\sqrt{2x-1}=3\left| 1-2x \right|$

a) Đặt $t=\sqrt{3-2x}$ để đưa phương trình trên về phương trình theo ẩn t.

b) Tìm t rồi sau đó tìm x.

7. $5-\sqrt{3-2x}=\left| 2x-3 \right|$ (PTNK BAN CD 2000-2001)

8. $20-\sqrt{3-2x}=\left| 2x-3 \right|$ (PTNK BAN CD 2004-2005)

9. $12-\sqrt{4-3x}=\left| 3x-4 \right|$ (PTNK BAN CD 2007-2008)

10. $\displaystyle\left| 1-2x \right|=6-\sqrt{2x-1}$

11. $2\left| 4-3x \right|+3\sqrt{3x-4}-2=0$

12. $3\left| 3x-1 \right|+8\sqrt{1-3x}=3$

13. $2\sqrt{\frac{6x-1}{2x}}=\frac{2x}{6x-1}+1$

a) Đặt $t=\sqrt{\frac{6x-1}{2x}}$ để đưa phương trình trên về phương trình theo ẩn t.

b) Tìm t rồi sau đó tìm x.

14. $2\sqrt{\frac{3x-1}{x}}=\frac{x}{3x-1}+1$ (PTNK BAN CD 2001-2002)

15. $2\sqrt{\frac{6x-1}{2x}}=\frac{2x}{6x-1}+1$

16. $2\sqrt{\frac{9x-1}{3x}}=\frac{3x}{9x-1}+1$

17. $2\sqrt{\frac{3x-3}{x}}=\frac{3x}{x-1}+1$

18. $\displaystyle\sqrt{\frac{6x-4}{x}}=\frac{x}{3x-2}+1$

19. $\displaystyle\sqrt{\frac{2x-1}{x}}+1+\sqrt{\frac{x}{2x-1}}=3\frac{x}{2x-1}$

20. $2\sqrt{\frac{x}{x-1}}-\sqrt{\frac{x-1}{x}}=2\frac{x-1}{x}+3$

21. $3\sqrt{\frac{2x}{x-1}}+4\sqrt{\frac{x-1}{2x}}=3\frac{x-1}{2x}+10$

22. $\displaystyle\sqrt{\frac{x}{3-2x}}+5\sqrt{\frac{3-2x}{x}}=4\frac{3-2x}{x}+5$

23. $2\sqrt{\frac{2x+1}{x}}-3\sqrt{\frac{x}{2x+1}}=4\frac{x}{2x+1}+7$

24. $\displaystyle\frac{2+\sqrt{19-2x}}{x}=1$

25. $\displaystyle\sqrt{x}-\frac{4}{\sqrt{2+x}}+\sqrt{2+x}=0$

26. $\displaystyle\sqrt{9-5x}=\sqrt{3-x}+\frac{6}{\sqrt{3-x}}$

27. $\displaystyle\sqrt{2-x}+\frac{4}{3+\sqrt{2-x}}=2$

28. $\displaystyle\frac{\left( 5-x \right)\sqrt{5-x}+\left( x-3 \right)\sqrt{x-3}}{\sqrt{5-x}+\sqrt{x-3}}=2$

Bài 4: Giải các phương trình sau:

1. $\displaystyle\sqrt{2x+3}+\sqrt{2x+2}=1$

2. $\displaystyle\sqrt{1-x}+\sqrt{4+x}=3$

3. $\displaystyle\sqrt{x+4}-\sqrt{2x-6}=1$ ( PTNK, AB, 2006-2007, Vòng 1)

4. $\displaystyle\sqrt{3x-5}+\sqrt{2x+3}=\sqrt{x+2}$ (HSG, Q. Tân Bình, 2003-2004)

5. $\displaystyle\sqrt{x-2}+\sqrt{x-1}=\sqrt{2x-3}$ ( LÊ HỒNG PHONG 2001-2002, Vòng 1)

6. $\displaystyle\sqrt{3x+7}-\sqrt{x+1}=2$

7. $\displaystyle\sqrt{x+3}+\sqrt{x-1}=2$

8. $\displaystyle\sqrt{x+5}+\sqrt{3-x}=4$

9. $\displaystyle\sqrt{{{x}^{2}}+9}-\sqrt{{{x}^{2}}+7}=2$

10. $\displaystyle\sqrt{2-{{x}^{2}}}+\sqrt{{{x}^{2}}+8}=4$

11. $\displaystyle\sqrt{x+3}-\sqrt{7-x}=\sqrt{2x-8}$

12. $\displaystyle\sqrt{2-x}=\sqrt{7-x}-\sqrt{-3-2x}$

13. $\displaystyle\sqrt{11-x}-\sqrt{x-1}=2$

14. $\displaystyle\sqrt{{{x}^{2}}+3x+2}-\sqrt{{{x}^{2}}+x+1}=1$

15. $\displaystyle\sqrt{5x-1}=\sqrt{3x-2}-\sqrt{2x-3}$

16. $\displaystyle\sqrt{5x-1}-\sqrt{x-1}=\sqrt{2x-4}$

17. $\displaystyle\sqrt{x+2}-\sqrt{2x-3}=\sqrt{3x-5}$ (HSG, Q.I,1999-2000)

18. $\displaystyle\sqrt{x+4}-\sqrt{1-x}=\sqrt{1-2x}$

19. $\displaystyle\sqrt{x+9}=5-\sqrt{2x+4}$

20. $\displaystyle\sqrt{3x+4}-\sqrt{2x+1}=\sqrt{x+3}$

21. $\displaystyle\sqrt{x+12}+\sqrt{x-6}-\sqrt{x+2}-\sqrt{x-4}=0$

22. $\displaystyle\sqrt{3x+6}+\sqrt{3x-3}-\sqrt{3x+1}-\sqrt{3x-2}=0$

23. $\displaystyle\sqrt{x+4}+\sqrt{x-5}-\sqrt{x-1}-\sqrt{x-4}=0$

24. $\displaystyle\sqrt{2x+6}+\sqrt{2x-3}-\sqrt{2x+1}-\sqrt{2x-2}=0$

25. $\displaystyle\sqrt{x+6}+\sqrt{x-3}-\sqrt{x+1}-\sqrt{x-2}=0$ (PTNK, AB, 2005-2006, Vòng 1)

26. $\displaystyle\left\{ \begin{array}{l}x-y=5\\\sqrt{2x+1}-\sqrt{y+2}=2\end{array} \right.$ ( PTNK, AB, 2005-2006, Vòng 1)

27. $\displaystyle\sqrt{x+\sqrt{x+11}}+\sqrt{x-\sqrt{x+11}}=4$

28. $\displaystyle\sqrt{x-\sqrt{x-2}}+\sqrt{x+\sqrt{x-2}}=3$

29. $\displaystyle\frac{2-\sqrt{x}}{2-x}=\sqrt{\frac{2}{2-x}}$

30. $\displaystyle\sqrt{\frac{20+x}{x}}+\sqrt{\frac{20-x}{x}}=\sqrt{6}$ ( đặt $t=\frac{20}{x}$ )

Bài 5: Giải các phương trình sau:

1. $\displaystyle\sqrt{2x+1}+\sqrt{x-3}=2\sqrt{x}$

2. $\displaystyle\sqrt{x+2}-\sqrt{2x-3}=\sqrt{4x-7}$

3. $\displaystyle\sqrt{x}+\sqrt{x-3}=\sqrt{3\left( x-1 \right)}$

4. $\displaystyle\sqrt{x-2}+\sqrt{4-x}=\sqrt{6-x}$

5. $\displaystyle\sqrt{x+5}=2\sqrt{x}-\sqrt{2x-7}$

6. $\displaystyle\sqrt{3x+12}-\sqrt{4x+13}=\sqrt{x+1}$

7. $\displaystyle\sqrt{9-x}-\sqrt{x+4}=\sqrt{3x+1}$

8. $\displaystyle\sqrt{3x+4}=2\sqrt{x}-\sqrt{x-4}$

9. $\displaystyle\sqrt{2x+5}=\sqrt{12x+25}-\sqrt{5x+6}$

10. $\displaystyle\sqrt{x+1}+\sqrt{x-1}=\sqrt{3x-1}$

11. $\displaystyle\sqrt{3x-5}+\sqrt{2x-3}=\sqrt{x+2}$

12. $\displaystyle\sqrt{{{x}^{2}}+9}-\sqrt{{{x}^{2}}-7}=2$

13. $\displaystyle\sqrt{{{x}^{2}}+5}+\sqrt{{{x}^{2}}-3}=4$

14. $\displaystyle\sqrt{{{x}^{2}}-3x+6}+\sqrt{{{x}^{2}}-3x+3}=3$

15. $\displaystyle\sqrt{3{{x}^{2}}-2x+15}+\sqrt{3{{x}^{2}}-2x+8}=7$

16. $\displaystyle\sqrt{3{{x}^{2}}+5x+8}-\sqrt{3{{x}^{2}}+5x+1}=1$

Bài 6: Giải các phương trình sau:

1. $\displaystyle\sqrt{x+6}+\sqrt{x-3}-\sqrt{x+1}-\sqrt{x-2}=0$

2. $\displaystyle\sqrt{{{x}^{2}}+x-5}+\sqrt{{{x}^{2}}+8x-4}=5$

3. $\displaystyle\sqrt{3{{x}^{2}}+6x+16}+\sqrt{{{x}^{2}}+2x}=2\sqrt{{{x}^{2}}+2x+4}$

4. $\displaystyle\sqrt{2{{x}^{2}}-9x+4}+3\sqrt{2x-1}=\sqrt{2{{x}^{2}}+21x-11}$

Bài 7: Giải các bất phương trình sau:

1. $\displaystyle\sqrt{3x-5{{x}^{2}}}\le 5x-2$ (PTNK, AB, 2006-2007, Vòng 1)

2. $\displaystyle\sqrt{5-x}\le 2x-7$

3. $\displaystyle\sqrt{x+1}<2x-1$ (PTNK, AB, 2002-2003, Vòng 1)

4. $\displaystyle\sqrt{{{x}^{2}}-x-12}<7-x$

5. $1-x+\sqrt{2{{x}^{2}}-3x-5}<0$

6. $\displaystyle\sqrt{{{x}^{2}}-3x-10}\le x-2$

7. $3\sqrt{-{{x}^{2}}+x+6}+2\left( 2x-1 \right)<0$

8. $\displaystyle\sqrt{3{{x}^{2}}+13x+4}+2-x\le 0$

9. $2\sqrt{3x+{{x}^{2}}}\le 2x-1$

10. $\displaystyle\sqrt{{{x}^{2}}-3x+3}\le 2x-1$

11. $\displaystyle\sqrt{{{x}^{2}}+3x-3}\le 2x-3$

12. $-6x+7\ge \sqrt{{{x}^{2}}-6x+7}$

13. $\displaystyle\sqrt{{{x}^{2}}-3x-10}\le 8-x$

14. $\displaystyle\sqrt{{{x}^{2}}+5x-6}<2x+3$

15. $\displaystyle\sqrt{x+3}<x+1$

16. $\displaystyle\sqrt{2x+12}<x+2$

17. $\displaystyle\ \sqrt{{{x}^{2}}-6x-40}\le 16-x$

18. $\displaystyle\sqrt{{{x}^{2}}+10x+4}\le 2x-4$

19. $3\sqrt{-{{x}^{2}}+3x+54}+4x-6<0$

20. $2-x+\sqrt{2{{x}^{2}}-6x-20}<0$

Bài 8: Giải các bất phương trình sau:

1. $\displaystyle\sqrt{3x-5{{x}^{2}}}\ge 5x-2$

2. $\displaystyle\sqrt{5-x}\ge 2x-7$

3. $\displaystyle\sqrt{x+1}\ge 2x-1$

4. $\displaystyle\sqrt{{{x}^{2}}-x-12}\ge 7-x$

5. $1-x+\sqrt{2{{x}^{2}}-3x-5}>0$

6. $\displaystyle\sqrt{{{x}^{2}}-3x-10}\ge x-2$

7. $3\sqrt{-{{x}^{2}}+x+6}+2\left( 2x-1 \right)>0$

8. $\displaystyle\sqrt{3{{x}^{2}}+13x+4}+2-x\ge 0$

9. $2\sqrt{3x+{{x}^{2}}}\ge 2x-1$

10. $\displaystyle\sqrt{{{x}^{2}}-3x+3}\ge 2x-1$

11. $\displaystyle\sqrt{{{x}^{2}}+3x-3}\le 2x-3$

12. $-4x+2\le 4\sqrt{{{x}^{2}}-6x+5}$

13. $\displaystyle\sqrt{{{x}^{2}}-3x-10}\ge 8-x$

14. $\displaystyle\sqrt{{{x}^{2}}+5x-6}>2x+3$

15. $\displaystyle\sqrt{x+3}>x+1$

16. $\displaystyle\sqrt{2x+12}>x+2$

17. $\displaystyle\ \sqrt{{{x}^{2}}-6x-40}\ge 16-x$

18. $\displaystyle\sqrt{{{x}^{2}}+10x+4}\ge 2x-4$

19. $3\sqrt{-{{x}^{2}}+3x+54}+4x-6>0$

20. $2-x+\sqrt{2{{x}^{2}}-6x-20}>0$

Đại số 9 - Tags: phương trình vô tỷ, toán 9