10类RD Sharma解决方案–第8章二次方程式–练习8.3 |套装1

通过分解求解以下二次方程

问题1.(x – 4)(x + 2)= 0

解决方案：

We have equation, 

(x – 4) (x + 2) = 0  

Implies that either x – 4 = 0 or x + 2 = 0

Therefore, roots of the equation are 4 or -2.

问题2.(2x + 3)(3x – 7)= 0

解决方案：

We have equation,

(2x + 3) (3x – 7) = 0  

Implies that either 2x + 3 = 0 or 3x – 7 = 0

Therefore, roots of the equation are -3/2 or 7/3.

问题3 ^3×2 – 14× – 5 = 0

解决方案：

We have equation,

3x² – 14x – 5 = 0

We can factorize this equation as 

3x² – 15x + x – 5 = 0  

3x (x – 5) -1 (x – 5) = 0

(3x – 1) (x – 5) = 0 

Therefore, roots of the equation are -1/3 or 5.

问题4. 9x ² – 3x – 2 = 0

解决方案：

We have equation,

9x² – 3x – 2 = 0

We can factorize this equation as

9x² – 6x + 3x – 2 = 0  

3x(3x – 2) + 1(3x – 2) = 0

(3x – 2) (3x + 1) = 0

Therefore, roots of the equation are 2/3 or -1/3.

问题5.(1 /(x – 1))–(1 /(x + 5))= 6/7，x≠1，-5。

解决方案：

We have equation,

 (1/(x – 1)) – (1/(x + 5)) = 6/7 

We can rewrite this equation as

(x + 5 – x + 1)/((x – 1) (x + 5)) = 6 / 7

1/((x – 1) (x + 5)) = 1/7, or

(x – 1) ( x + 5) = 7

x² + 4x – 12 = 0

We can factorize this equation as

x² + 6x – 2x – 12 = 0

x (x + 6) – 2 (x + 6) = 0

(x – 2) (x + 6) = 0

Therefore, roots of the equation are 2 or -6.

问题6. 6x ² + 11x + 3 = 0

解决方案：

We have equation,

6x² + 11x + 3 = 0

We can factorize this equation as

6x² + 9x + 2x + 3 = 0  

3x (2x + 3) +1 (2x + 3) = 0

(2x + 3) (3x + 1) = 0

Therefore, roots of the equation are -3/2 or -1/3.

问题7. 5X ² – 3× – 2 = 0

解决方案：

We have equation,

5x² – 3x – 2 = 0

We can factorize this equation as

5x² – 5x + 2x – 2 = 0  

5x (x – 1) + 2(x – 1) = 0

(x – 1) (5x + 2) = 0

Therefore, roots of the equation are -2/5 or -1.

问题8. 48x ² – 13x – 1 = 0。

解决方案：

We have equation,

48x² – 13x – 1 = 0

We can factorize this equation as

48x² – 16x + 3x – 1 = 0  

16x (3x – 1) + 1 (3x – 1) = 0

(3x – 1) (16x + 1) = 0

Therefore, roots of the equation are -1/16 or 1/3.

问题9. 3x ² = -11x – 10。

解决方案：

We have equation,

3x² + 11x +10 = 0

We can factorize this equation as

3x² + 6x + 5x + 10 = 0  

3x (x + 2) + 5 (x + 2) = 0

(3x + 5) (x + 2) = 0

Therefore, roots of the equation are -2 or -5 / 3.

问题10.25x(x + 1)= -4。

解决方案：

We have equation,

25x² + 25x + 4 = 0

We can factorize this equation as

25x² + 20x + 5x + 4 = 0  

5x (5x + 4) + 1 (5x + 4) = 0

(5x + 4) (5x + 1) = 0

Therefore, roots of the equation are -1/5 or -4/5.

问题11。16x–(10 / x)= 27

解决方案：

We have equation 16x² -27x -10 = 0

We can factorize this equation as:

16x² – 32x + 5x – 10 = 0

16x (x – 2) + 5 (x – 2) = 0

(16x + 5) (x – 2) = 0

Therefore, the roots of equations are 2 or -5/16.

问题12.(1 / x)–(1 /(x – 2))= 3，x≠0，2

解决方案：

We have equation 3x² – 6x + 2 = 0

Here, a = 3, b = -6 and c = 2

Since,

Discriminant (D) = b² – 4ac and x = (-b ± √D)/2a

Therefore, 

D = 36 – 24 = 12, and

x  = (-(-6) ± √12)/6

x = (6 ± 2√3)/6

x = (3 ± √3)/3

Therefore, the roots of equations are (3 + √3)/3 or (3 – √3)/3.

问题13. x –(1 / x)= 3，x≠0

解决方案：

 We have equation x² – 3x  -1 = 0

Here, a = 1, b = -3 and c = -1

Since,

Discriminant (D) = b² – 4ac and x = (-b ± √D)/2a

Therefore,

D = 9 + 4 = 13, and

x = (-(-3) ± √13)/2

x = (3 ± √13)/2

Therefore, the roots of equations are (3 + √13)/2 or (3 – √13)/2.

问题14.(1 /(x + 4))–(1 /(x – 7))= 11/30，x≠4，7

解决方案：

 We have equation,

(1/(x + 4)) – (1/(x – 7)) = 11/30

(x – 7 – x – 4)/((x + 4) (x – 7)) = 11/30

-11/((x – 4) (x – 7)) = 11/30

-1/((x – 4) (x – 7)) = 1/30

 x² – 3x + 2 = 0

We can factorize this equation as:

x² – 2x – x + 2 = 0

x (x – 2) – 1 (x – 2) = 0

(x -1) (x – 2) = 0

Therefore, the roots of equations are 2 or 1.

问题15((1 /(x – 3))+(2 /(x – 2))= 8 / x，x≠0，2，3

解决方案：

 We have equation,

(1/(x – 3)) + (2/(x – 2)) = 8/x

(x – 2 + 2x – 6)/((x – 3) (x – 2)) = 8/x

(3x – 8)/((x – 3) (x – 2)) = 8/x 

8 ((x – 3) (x – 2)) = x(3x – 8)

5x² – 32x + 48 = 0

We can factorize this equation as:

5x² – 20x -12x + 48 = 0

5x (x – 4) -12(x – 4) = 0

(5x – 12) (x – 4) = 0

Therefore, the roots of equations are 12/5 or 4.

问题16. a ² x ² – 3abx + 2b ² = 0

解决方案：

 We have equation,

a²x² – 3abx + 2b² = 0

We can factorize this equation as:

a²x² – 2abx – abx + 2b² = 0

ax (ax – 2b) – b (ax – 2b) = 0

(ax – b) (ax – 2b) = 0

Therefore, the roots of the equation are b/a or 2b/a. 

问题17. 9x ² – 6b ² x –(a ⁴ – b ⁴ )= 0

解决方案：

 We have equation,

9x² – 6b²x – (a⁴ – b⁴) = 0

We can factorize this equation as:

(3x)² – 6b²x + (b²)² – a⁴ = 0

(3x – b²)² – (a²)² = 0

(3x – b² + a²) (3x – b² – a²) = 0

Therefore, the roots of the equation are (b² – a²)/3 or (b² + a²)/3. 

问题^{18。4x 2} + 4bx –(a ² – b ² )= 0

解决方案：

We have equation,

4x² + 4bx – (a² – b²) = 0

Dividing by 4

x² + bx – ((a² – b²)/4) = 0

x² + bx – (((a – b)/2) ((a + b)/2)) = 0

We can write b as :

b = ((a + b)/2) – ((a – b)/2)

Therefore,

x² + ((a + b)/2) – ((a – b)/2) x – (((a – b)/2) ((a + b)/2)) = 0

x (x + ((a + b)/2)) – ((a – b)/2) (x + (a + b)/2) = 0

(x + ((a + b)/2)) (x – ((a – b)/2)) = 0

Therefore, the roots of the equation are (a – b)/2 or (-a – b)/2. 

问题19. ax ² +(4a ² – 3b)x – 12ab = 0。

解决方案：

We have equation,

 ax² + 4a²x – 3bx – 12ab = 0

ax (x + 4a) – 3b (x – 4a) = 0

(ax – 3b) (x + 4a) = 0

Therefore, the roots of the equation are 3b/a or -4a.

问题20. 2x ² + ax – a ² = 0。

解决方案：

We have equation,

2x² + ax – a² = 0

2x² + 2ax – ax – a² = 0

2x (x + a) – a (x + a) = 0

(2x – a) (x + a) = 0

Therefore, the roots of the equation are a/2 or -a.