Vở Bài Tập Toán 9-Hki-2023_2024_Sưu Tầm.pdf

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Chương

1
Bài 1. CĂN BẬC HAI SỐ HỌC
A. KIẾN THỨC TRỌNG TÂM
1. Căn bậc hai số học
 Với số dương a , số a được gọi là căn bậc hai số học của a .
 Số 0 cũng được gọi là căn bậc hai số học của 0.
x  0

 Với số a khơng âm, ta có a  x   2
.
x  a

2. So sánh hai căn bậc hai số học
 Với hai số a và b không âm, ta có a  b  a  b .
B. CÁC DẠNG BÀI TẬP VÀ PHƯƠNG PHÁP GIẢI
Dạng 1: Tìm căn bậc hai, căn bậc hai số học của một số



x  0
Dựa vào định nghĩa căn bậc hai số học của một số a  x  
 2

x  a.




Ví dụ 1. Tìm căn bậc hai số học của mỗi số sau rồi suy ra căn bậc hai của chúng.

a) 0 ;

b) 81 ;

e) 0,25 ;

f)

169
;
49

c) 196 ;
g)

36
;
121

d) 4, 41 ;
h) 3

6
.
25

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256
;
225

h) 1

15
.
49

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Dạng 2: Tính giá trị của biểu thức chứa căn bậc hai


Sử dụng kiến thức: với a  0 , ta có

a 2  a;

 a

2

a.

Ví dụ 3: Tính:
a) 16 ;

0, 81 ;

b)

c)

324
;
289

d)

625
.
64

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Ví dụ 4: Tính:
a) 25 ;

0,16 ;

b)

c)

25
;
81

d)

64
.
49

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Ví dụ 5: Tính:

 75  ;
2

a)

b)





2

0, 4 ;

2

 4 

 ;
c) 
 81 

2

 19 

 .
d) 
 16 

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2

Ví dụ 6: Tính:

 19  ;
2

a)

b)



2

 10 

 ;
c) 
 9 



2

0,16 ;

2

 27 

 50
 1.
9
4

ĐS: 22 .

ĐS: 7 . b) 2  2
ĐS: 13 . d) 3

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Ví dụ 8: Tính giá trị của các biểu thức sau:
a) 0, 5 64  2 25 ;

c)

1
2
9
25 ;
3
5

ĐS: 6 . b) 10  1, 69  5  1
ĐS: 1 . d) 9

11
;
25

121 3 196

 27 .
9
2 9

ĐS: 19 .
ĐS: 1 .

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3

Dạng 3: Tìm giá trị của x thỏa mãn biểu thức cho trước



x 2  a 2  x  a hoặc x  a .



Với a  0 thì x 2  a  x  a hoặc x   a .

Ví dụ 9: Tìm x , biết:
a) x 2  289 ;

ĐS: x  17 . b) 25x 2  16 ;

c) 0, 49x 2  2, 56 ;

ĐS: x  

16
. d) 9x 2  10  0 .
7

4
5

ĐS: x   .
ĐS: Vơ nghiệm.

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ĐS: x   31 .

d) 27x 2  6  0 .

ĐS: x  

2
.
3

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Ví dụ 12: Tìm x , biết:
a) x 2  2 ;
c) 64x 2  13 ;

ĐS: x   2 .
ĐS: x  

13
.
8

b) x 2  15  0 ;
d) 49x 2  26  0 .

ĐS: x   15 .
ĐS: x  

26
.
7

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5

Ví dụ 13: Tìm x khơng âm, biết:
a) x  21 ;
c)





2

x 1  4;

ĐS: x  441 .
ĐS: x  1 .

b) 2 x  1 ;
d)

x 1  2.

ĐS: Vơ nghiệm.
ĐS: x  9 .

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ĐS: Vô nghiệm.
ĐS: x  9 .

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6

Dạng 4: So sánh các căn bậc hai số học


Sử dụng định lý: với a, b  0 : a  b  a  b .

Ví dụ 15: So sánh:
a) 6 và

37 ;

b) 4 và

c) 10  3 và 6 ;

37  2 ;

d) 4 và 26  1 .

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Ví dụ 16: So sánh:
a) 6 và

41 ;

b) 3 2 và 5 ;

c) 5  1 và 3 ;

d) 4 và 17  2 .

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ĐS: 0  x 

9
.
50

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Ví dụ 19: Chứng minh rằng với x  0 thì
a) x  3  3 ;
c)

3
x 1

 3;

b) 3  x  3 ;
d) 1 

5

3
 .
2
x 2

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8

Ví dụ 20: Chứng minh rằng với x  0 thì
b) 2  x  2 ;

a) x  2  2 ;
c)

4
x 2

 2;

d) 1 

1
x 2



1
.
2

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C. BÀI TẬP VẬN DỤNG

Bài 1: Tìm căn bậc hai số học của mỗi số sau rồi suy ra căn bậc hai số học của chúng.
a) 0 ;

b) 64 ;

e) 0, 36 ;

f)

169
;
324

c) 289 ;
g)

49
;
144

d) 2, 56 ;
h) 2

14
.
25

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Bài 2: Tính:
a)

361 ;

b)

0, 01 ;

c)

64
;
25

d)

25
.
9

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9

Bài 3: Tính:

 23  ;
2

a)

b)



2

 9 

 ;
c) 
 16 



2

1,2 ;

2

 25 


 .
d) 
 4 

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Bài 4: Thực hiện phép tính:
a) 3 4  8 9  15 16 ;

c)

2
3
9
36  19 ;
3
2

ĐS: 30 .
ĐS: 12 .

b) 5 0,16  3 0, 04 ;

d) 11

81
1
3
 1.
121
9

ĐS:

13
.
5

ĐS: 9 .

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Bài 5: Tìm x , biết
a) x 2  400 ;

ĐS: 20 .

c) 0,16x 2  0, 09 ;

ĐS:  .

3
4

b) 75x 2  48 ;
d) 27x 2  10  0 .

4
5

ĐS:  .
ĐS: Vô nghiệm.

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10

Bài 6: Tìm x , biết:
ĐS:  11 .

a) x 2  11 ;

ĐS: 

c) 9x 2  17 ;

17
.
3

b) x 2  7  0 ;

ĐS:  7 .

d) 12x 2  21  0 .

ĐS: 

7
.
2

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d) 1  x  3 .

ĐS: 16 .

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11

Bài 8: So sánh:
a) 7 và

c) 15  4 và 8 ;

b) 2 5 và 4 ;

41 ;

d) 3 và 17  1 .

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Bài 9: Tìm x khơng âm, biết:
ĐS: 0  x  9 .

a) x  3 ;
c)

3x  2  5 ;

ĐS: x 

49
.
3

b)

ĐS: 0  x  0, 09 .

4x  0, 6 ;
3
4

ĐS: 0  x 

d) 2  x  .

25
.
16

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Bài 10: Chứng minh rằng với x  0 thì

a) x  3  3 ;
c) 1 

2
x 1

 1 ;

b) 2 x  1  1 ;
d) 0 

12

7
x 3



7
.
3

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--- HẾT ---

13

Chương

1
Bài 2. CĂN THỨC BẬC HAI.

HẰNG ĐẲNG THỨC BẬC HAI
A. KIẾN THỨC TRỌNG TÂM
 Với A là biểu thức đại số, ta gọi

A là căn thức bậc hai của A, còn A được gọi là biểu thức

lấy căn hoặc biểu thức dưới dấu căn.


A xác định (hay có nghĩa) khi và chỉ khi A 0 .


 A neáu A  0
 Hằng đẳng thức A  A  


A neáu A  0.



B. CÁC DẠNG BÀI TẬP VÀ PHƯƠNG PHÁP GIẢI
2

Dạng 1: Tìm giá trị của biểu thức chứa căn bậc hai



 A neáu A  0
A2  A  


A nếu A  0.



Sử dụng hằng đẳng thức

Ví dụ 1: Tính:
a) 25 ;

b)

2, 5

2

;

c)

81
;
100

d)

 121 
 .
 
 49 

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Ví dụ 2: Tính:
a) 132 ;

b)

2

2

;

c)

64
;
25

 36 
.
 169 

d)  

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14

Ví dụ 3: Rút gọn các biểu thức sau:
a)



c)

42 3 ;

3 2



2

;





2

ĐS: 3  2 .

b)

ĐS:

d) 7  4 3 .

3  1.

11  3 ;

ĐS: 11  3 .
ĐS: 2  3 .

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ĐS: 7  3 .

ĐS: 5  1 .

d)

82 7 .

ĐS: 1  7 .

2

7 3 ;

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Ví dụ 5: Thực hiện các phép tính:
a) 196  25  5 81 ;

c)





10  3

2

 10 ;





ĐS: 25 .

b) 32 : 16  289  49 ;

ĐS: 3 .

d)

15



5 7



2



8  2 7  .

ĐS: 175 .
ĐS: 6 .

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Ví dụ 6: Thực hiện các phép tính:
a) 64  25  10 36 ;



c)



2

7 1  7 ;





ĐS: 100 .

b) 81 : 9  169  225 ;

ĐS: 1 .

d)





2

3 1  42 3 .

ĐS: 600 .

ĐS: 2 .

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Ví dụ 7: Chứng minh:



a) 3  7
c)



2

 16  6 7 ;

b) 11  20  6 11  3 ;

41  12 5  41  12 5  2 5

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16

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Ví dụ 8: Chứng minh:



a) 1  2



2

 32 2;

b) 6  2 5  5  1 ;

c) 7  4 3  7  4 3  2 3 .

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Dạng 2: Tìm điều kiện để biểu thức chứa căn bậc hai có nghĩa


A xác định (hay có nghĩa) khi và chỉ khi A 0 .

Ví dụ 9: Với giá trị nào của a thì mỗi căn thức sau có nghĩa:
a)

72a ;

c) 19  4a ;

ĐS: a  0 .
ĐS: a 

19
.
4

b)

13
;
3a

d) 27  6a .

17

ĐS: a  0 .
9
2

ĐS: a  .

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17
.
5

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Ví dụ 11: Với giá trị nào của x thì mỗi căn thức sau có nghĩa:
a)

15
;
x 2

ĐS: x  2 .

c)

10  30x
;
3x 2  1

ĐS: x  .

1
3

b)

17
;
12  x

d)

4x  2
.
x  4x  5
2

ĐS: x  12 .
ĐS: x 

1
.
2

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22
.
5

d)

x 2
.
x  2x  3

ĐS: x  2 .

2

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ĐS: 2a 2 .

d) 9a 6  2a 3 với a  0 .

ĐS: 5a 3 .

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19

Ví dụ 14: Rút gọn các biểu thức sau:
a) 2 a 2 với a  0 ;

ĐS: 2a .

b) 16a 2  4a với a  0 ;

ĐS: 0 .

c) a 4  4a 2 ;

ĐS: 3a 2 .

d) a 6  a 3 với a  0 .

ĐS: 0 .

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Ví dụ 15: Rút gọn các biểu thức sau:

a  4

ĐS: a  4 .

b)

5  a 

c) a 2  6a  9 với a  3 ; ĐS: a  3 .

d)

4a 2  4a  1  2a với a 

a)

2

với a  4 ;

2

ĐS: 5  3a .

 4a với a  5 ;

1
.
2

ĐS: 1 .

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Ví dụ 16: Rút gọn các biểu thức sau:
a)

a  1

2

với a  1 ;

ĐS: a  1 .

c) a 2  2a  1 với a  1 ; ĐS: a  1 .

b)

2  a 

2

ĐS: 2 .

 a với a  2 ;

1
3

d) 9a 2  6a  1  3a với a  .

ĐS: 1 .

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a) x 2  3 ;

b) 9x 2  5 ;

c) x 2  2 2x  2 ;

d) 4x 2  4 3x  3 .

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Dạng 5: Giải phương trình









Bước 1: Tìm điều kiện xác định.
Bước 2: Biến đổi hai vế về các phương trình đã biết cách giải.
Bước 3: Đối chiếu điều kiện rồi kết luận nghiệm của phương trình.
Các phép biến đổi thường gặp

B  0
A2  B  



A  B



B  0
A2  B  


| A | B


A2  B 2 | A || B | A  B .

Ví dụ 19: Giải các phương trình sau:
a) x 2  5  0 ;

ĐS: x   5 .

b) 4x 2  2  0 ;

c) x 2  2 5x  5  0 ;

ĐS: x   5 .

d) 4x 2  4 2x  2  0 .

ĐS: x  

1
.
2

ĐS: x 

2
.
2

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21

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Ví dụ 20: Giải các phương trình sau:
3
.

4

a) x 2  2  0 ;

ĐS: x   2 .

b) 4x 2  3  0 ;

ĐS: x 

c) x 2  2 3x  3  0 ;

ĐS: x   3 .

d) x 2  2 2x  2  0 .

ĐS: x  2 .

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Ví dụ 21: Giải các phương trình sau:
a) x 2  8 ;

c)

4x 2  19  0 ;

ĐS: x  8 .
ĐS: x  

19
.
2

b) 9x 2  10 ;

d)

49x 2 | 14 | .

ĐS: x  

10
.
3

ĐS: x  2 .

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1
4

ĐS: x   .
ĐS: x  2 .

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Ví dụ 23: Giải các phương trình sau:
a)

x  2

2

 3;

ĐS: S  {1;5} .

b) 25  10x  x 2  1 ;

ĐS: S  {4;6} .

ĐS: S   .

d) 9x 2  6x  1  x ;

ĐS: S   .

c) x 2  4x  4  1  x ;
e) x  2 x  1  0 ;

ĐS: x  1 .

f) x  2 x  3  0 .

ĐS: x  9 .

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23
23

c) x 2  2x  1  2  x ;

ĐS: x 

e) x  4 x  4  0 ;

ĐS: x  4 .

b) 9  6x  x 2  1 ;

x  4
ĐS: 
.

d) x 2  6x  9  x  1 ;

ĐS: S   .

f) x  4 x  5  0 .

ĐS: x  25 .

x  2

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