Chapter 4 - Quadratic Equations, RD Sharma Solutions - (Part-1) | RD Sharma Solutions for Class 10 Mathematics PDF Download

Page No 4.19

Q.1. Solve the following quadratic equations by factorization: 
(x − 4) (x + 2) = 0
Ans.

We have been given,
(x- 4)(x - 2) = 0
Therefore,

(x - 4) = 0
x = 4

or,

(x+2) = 0
x = -2

Therefore,

x = 4 or x = -2

Q.2. Solve the following quadratic equations by factorization:
(2x + 3)(3x − 7) = 0
Ans.
We have been given,
(2x + 3)(3x-7) = 0
Therefore,

(2x+ 3) = 0
2x =-3
x = -3/2

or,

(3x-7) = 0
3x = 7
x = 7/3

Therefore,
x = -3/2 or x = 7/3

Q.3. Solve the following quadratic equations by factorization: 3x − 14x − 5 = 0
Ans.
We have been given

3x² - 14x - 5 = 0
3x² - 15x + x - 5 = 0
3x(x - 5) +1(x - 5) = 0
(3x + 1)(x - 5) = 0

Therefore,

3x+1 = 0
3x = -1
x = -1/3

or,

x-5 = 0
x = 5
Hence, x = -1/3 or x = 5

Q.4. Solve the following quadratic equations by factorization:
9x² − 3x − 2 = 0
Ans.
We have been given,
9x² - 3x - 2 = 0

9x² - 6x + 3x - 2 = 0
3x(3x-2)+1(3x-2) = 0
(3x+1)(3x-2) = 0 

Therefore,

3x+1 = 0
3x = -1
x = -1/3

or,

3x-2 = 0
3x = 2

Hence,
x = -1/3 or x = 2/3

Q.5. Solve the following quadratic equations by factorization:

Ans.
We have been given 

x² _{+ 4x-12 = 0}

x²+6x - 2x - 12 = 0
x(x+6)-2(x+6) = 0
(x-2)(x+6) = 0

Therefore,

x-2 = 0
x= 2

or,

x+6 = 0
x = -6

Hence,
x=2 or x = -6

Q.6. Solve the following quadratic equations by factorization:
6x² + 11x + 3 = 0
ANS.
We have been given
6x²+11x+3 = 0

6x²+9x-2x +3 = 0
3x(2x+3)+1(2x+3) = 0
(2x+3)(3x+1) = 0
2x+3 = 0
x = -3/2

or,

3x+1 = 0
x = -1/3

Hence,
x = -3/2 or x = -1/3

Q.7. Solve the following quadratic equations by factorization:
5x² − 3x − 2 = 0
Ans.
We have been given
5x² − 3x − 2 = 0

5x²-5x+2x-2 = 0
5x(x-1)+2(x-1) = 0
(5x+2)(x-1) = 0

Therefore,

5x+2 = 0
5x = -2
x = -2/5

or,

x - 1 = 0
x = 1

Hence,
x = -2/5 or x = 1

Q.8. Solve the following quadratic equations by factorization:
48x² − 13x − 1 = 0
ANS.
We have been given
48x² − 13x − 1 = 0

48x² _{- 16x + 3x - 1 = 0}
16x(3x-1)+1(3x-1) = 0
(16x+1)(3x-1) = 0

Therefore,

16x+1 = 0
16x = -1
x = -1/16

or

3x-1 = 0
3x = 1
x = 1/3

Hence,
x = -1/16 or x = 1/3

Q.9. Solve the following quadratic equations by factorization:
3x² = −11x − 10
ANS.
We have been given
3x² = −11x − 10

3x² _{+ 11x + 10 = 0}
3x²+6x+5x +10 = 0
3x(x+2)+5 (x+2)=0
(3x+5)(x+2)=0

Therefore,

3x+5 = 0
3x = -5
x = -5/3

or,

x+2 =0
x = -2

Hence,
x = -5/3 or x = -2

Q.10. Solve the following quadratic equations by factorization:
25x (x + 1) = −4
ANS.
We have been given
25x (x + 1) = −4

25x²+25x+4 = 0
25x²+20x + 5x + 4 = 0
5x(5x+4)+1(5x+4) = 0
(5x+1)(5x+4) = 0

Therefore,

5x+1 = 0
5x = -1
x = -1/5

or,

5x+4 = 0
5x = -4
x = -4/5

Hence,
x = -1/5 or x = -4/5

Q.11. Solve the following quadratic equations by factorization:

Ans.

16x² - 10 = 27x
16x² _{- 27x - 10 = 0}
16x² _{-32x + 5x - 10 = 0
16x(x-2) + 5(x - 2) = 0
(16x+5)(x-2) = 0
16x + 5 = 0 or x = 2 = 0}

_{x = -5/16 or x = 2}

_{Hence, the factor are 2 and -5/16}

Q.12. Solve the following quadratic equations by factorization:

Ans.
We have been given

-2x = 3x² _{- 6x
3x}² _{- 6x + 2 = 0}
3x² - (3 + √3)x-(3-√3)x+3-√3+√3-1=0

(√3x - √3+1)(√3x - √3-1)=0

Therefore,

√3x - √3+1=0
√3x = √3-1
x = (√3-1)/√3

or,

√3x - √3-1
√3x = √3+1
x = (√3+1)/√3

Hence,
x = (√3-1)/√3 or x = (√3+1)/√3

Q.13. Solve the following quadratic equations by factorization:

Ans.
We have been given

x - 1/x = 3

x²-1 = 3x
x² - 3x -1 = 0

Therefore,

or,

Hence

Q.14. Solve the following quadratic equations by factorization:

Ans.
We have been given

-30x = x² - 3x - 28
 -3x + 2 = 0
x² _{- 2x - x + 2 = 0
x(x-2)-1(x-2) = 0
(x-1)(x-2) = 0}

_Therefore,

x-1=0
x=1

or,

x-2 = 9
x = 2

Hence, x=1 or x=2

Q.15. Solve the following quadratic equations by factorization:

Ans.

⇒ x(3x−8) = 8(x²−5x+6)

⇒ 3x²−8x=8x²−40x+48
⇒5x²−32x+48=0
⇒5x²−20x−12x+48=0
⇒5x(x−4)−12(x−4)=0
⇒(5x−12)(x−4)=0
⇒5x−12=0 or x−4=0
⇒x=125 or x=4

Hence, the factors are 4 and 12/5

Q.16. Solve the following quadratic equations by factorization:
a²x² - 3abx + 2b² = 0
Ans.
We have been given

a²x² - 3abx + 2b² = 0
a²x² - 2abx - abx+2b² = 0
_{ax(ax-2b) - b(ax-2b) = 0
(ax-b)(ax - 2b) = 0}

_Therefore, 

ax - b = 0
ax= b
x = b/a

or,

ax - 2b = 0
ax = 2b
x = 2b/a

Hence,
x = b/a or x = 2b/a

Q.17. Solve the following quadratic equations by factorization:
9x²−6b²x−(a⁴−b⁴)=0
Ans.
9x²−6b²x−(a⁴−b⁴)=0

⇒9x²−6b²x−(a²−b²)(a²+b²)=0
⇒9x²+3(a²−b²)x−3(a²+b²)x−(a²−b²)(a²+b²)=0
⇒3x[3x+(a²−b²)]−(a²+b²)[3x+(a²−b²)]=0
⇒[3x−(a²+b²)][3x+(a²−b²)]=0
⇒3x−(a²+b²)=0 or 3x+(a²−b²)=0

Hence, the factor are

Q.18. Solve the following quadratic equations by factorization:
4x²+4bx−(a²−b²)=0
Ans.
We have been given

4x² + 4bx - (a² - b²) = 0
4x² + 2(a + b)x-2(a-b)x-(a² - b²) = 0
2x(2x+a+b) -(a-b)(2x+a+b) = 0
(2x-(a-b))(2x+a+b) = 0

Therefore,

2x-(a-b) = 0
2x = a-b
x = (a-b)/2

or,

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

Hence,

x = (a-b)/2 or x = -(a+b)/2

Q.19. Solve the following quadratic equations by factorization:
ax²+(4a²−3b)x−12ab=0
Ans.
We have been given

ax² + (4a² -3b)x - 12ab = 0
ax² +4a²x -3bx - 12ab = 0
ax(x+4a) - 3b(x+4a) = 0
(ax-3b)(x+4a) = 0

Therefore,

ax-3b = 0
ax = 3b
x = 3b/2

or,

x+4a = 0
x = -4a

Hence,
x = 3b/2 or x = -4a

Q.20. Solve the following quadratic equations by factorization:
2x²+ax−a²=0
Ans.

2x²+ax−a²=0
⇒ 2x²+2ax−ax−a²=0
⇒ 2x(x+a)−a(x+a)=0
⇒ (2x−a)(x+a)=0
⇒ 2x−a=0 or x+a=0
⇒ x = a/2 or x = -a

Hence, the factor are a/2 and -a

Q.21. Solve the following quadratic equations by factorization:

Ans.

⇒ (16−x)(x+1)=15x
⇒ 16x+16−x²−x=15x
⇒ −x²+16+15x=15x
⇒ −x²+16=0
⇒ x²−16=0
⇒ (x−4)(x+4)=0
⇒ x−4=0 or x+4=0
⇒ x=4 or x=−4

Hence, the factors are 4 and −4.

Page No 4.20

Q.22. Solve the following quadratic equations by factorization:

Ans.
We have been given

 
2x² - 3x+6x - 9 = 3x² - 7x+6x - 14
x² - 4x - 5 = 0
x² - 5x + x - 5 = 0
x(x-5)+1(x-5) = 0
(x+1)(x-5) = 0

Therefore,

x+1 = 0
x = -1

or,

x-5= 0
x = 5

Hence,
x = -1 or x = 5

Q.23. Solve the following quadratic equations by factorization:

Ans.
We have been given

 

6x² - 18x + 6x² - 24x - 15x + 60 = 25x² - 175x+300
13x² - 118x + 240 = 0
13x² -78x - 40x + 240
13x(x-6) - 40(x-6) = 0
(x-6)(13x-40) = 0

Therefore,

x-6= 0
x = 6

or,

13x - 40 = 0
13x = 40
x = 40/13

Hence,
x = 6 or x = 40/13

Q.24. Solve the following quadratic equations by factorization:

Ans.
We have been given

4(x²+3x-x+x²+2-2x) = 17(x²-2x)
9x² - 34x - 8 = 0
9x² - 36x + 2x - 8 = 0
9x(x-4)+2(x-4)=0
(9x+2)(x-4) = 0

Therefore,

9x+2 = 0
9x = -2
x = -2/9

or,

x-4 = 0
x= 4

Hence,
x = -2/9 or x = 4

Q.25. Solve the following quadratic equations by factorization:

Ans.
We have been given

7(x²+9-6x-9- 6x) = 48(x²-9)
48x² + 84x - 432 = 0
4x² + 7x - 36 = 0

Therefore,

4x²+16x-9x - 36 = 0
4x(x+4) - 9(x+4) = 0
(4x - 9) (x+4) = 0

Therefore,
4x-9 =0
4x = 9
x = 9/4

or,

x+4 = 0
x = -4

Hence,
x = 9/4 or x = -4

Q.26. Solve the following quadratic equations by factorization:

Ans.
We have been give

x²-x+2x² - 4x = 6 (x²-x-2x+2)
3x² - 13x + 12 = 0
Therefore,

3x² - 9x - 4x + 12 = 0
3x(x-3) -4(x-3) = 0
(3x-4)(x-3) = 0

Therefore,

3x - 4 = 0
3x = 4
x = 4/3

or,

x-3 = 0
x = 3

Hence,
x = 4/3 or x = 3

Q.27. Solve the following quadratic equations by factorization:

Ans.
We have been given

6(x²+1+2x-x²-1+2x) = 5(x²-1)
5x² - 24x - 5 = 0
5x² - 25x + x - 5 = 0
5x(x-5)+1(x-5) = 0
(5x+1) (x-5)=0

Therefore, 

5x+1 = 0
5x = -1
x = -1/5

or,

x-5 = 0
x= 5

Hence,
x = -1/5 or x = 5

Q.28. Solve the following quadratic equations by factorization:

Ans.

We have been given

2(x²+1-2x+4x²+1+4x) = 5(2x² - x-1)
10x² + 4x+4 = 10x²- 5x - 5
9x+9 = 0

Therefore,

9x = -9
x = -9/9
x = -1

Hence,
x = -1

Q.29. Solve the following quadratic equations by factorization:

Ans.

⇒(4−3x)(2x+3) = 5x
⇒8x+12−6x²−9x = 5x
⇒−6x²−6x+12=0
⇒x²+x−2 = 0
⇒x²+2x−x−2 = 0
⇒x(x+2)−1(x+2) = 0
⇒(x−1)(x+2) = 0
⇒x−1=0 or x+2 = 0
⇒x = 1 or x = −2

Hence, the factors are 1 and −2.

Q.30. Solve the following quadratic equations by factorization:

Ans.

⇒ 3(2x²−22x+58) = 10(x²−12x+35)
⇒ 6x²−66x+174 = 10x²−120x+350
⇒ 4x²−54x+176 = 0
⇒ 2x²−27x+88 = 0
⇒ 2x²−11x−16x+88 = 0
⇒ x(2x−11)−8(2x−11) = 0
⇒ (x−8)(2x−11) = 0
⇒ x−8=0 or 2x−11=0
⇒ x = 8 or x = 11/2

Hence, the factors are 8 and  11/2.

Q.31. Solve the following quadratic equations by factorization:

Ans.

⇒ (7x−24)(x−5) = (3x−9)(x−4)
⇒ 7x²−59x+120 = 3x²−21x+36
⇒ 4x²−38x+84 = 0
⇒ 2x²−19x+42 = 0
⇒ 2x²−12x−7x+42 = 0
⇒ 2x(x−6)−7(x−6) = 0
⇒ (2x−7)(x−6) = 0
⇒ 2x−7=0 or x−6 = 0
⇒ x=72 or x = 6

Hence, the factors are 6 and 7/2

Q.32. Solve the following quadratic equations by factorization: 

Ans.

⇒ 16x = 75−3x²
⇒ 3x²+16x−75 = 0
⇒ 3x²+25x−9x−75 = 0
⇒ x(3x+25)−3(3x+25) = 0
⇒ (x−3)(3x+25) = 0
⇒ x−3 = 0 or 3x+25 = 0
⇒ x = 3 or x = -25/3

Hence, the factors are 3 and -25/3

Q.33. Solve the following quadratic equations by factorization:

Ans.

⇒ (5−x)(3x−1) = 2(2x+2)
⇒ 15x−5−3x²+x = 4x+4
⇒ −3x²+16x−5−4x−4 = 0
⇒ −3x²+12x−9 = 0
⇒ 3x²−12x+9 = 0
⇒ x²−4x+3 = 0
⇒ x²−3x−x+3 = 0
⇒ x(x−3)−1(x−3) = 0
⇒ (x−1)(x−3) = 0
⇒ x−1=0 or x−3 = 0
⇒ x=1 or x=3

Hence, the factors are 3 and 1.

Q.34. Solve the following quadratic equations by factorization:

Ans.

⇒ (7x+1)(4x−1) = 29(x²−1)
⇒ 28x²−7x+4x−1 = 29x²−29
⇒ 29x²−28x²+3x−28 = 0
⇒ x²+3x−28 = 0
⇒ x²+7x−4x−28 = 0
⇒ x(x+7)−4(x+7) = 0
⇒ (x−4)(x+7) = 0
⇒ x−4=0 or x+7 = 0
⇒ x = 4 or x = −7

Hence, the factors are 4 and −7.

Q.35. Solve the following quadratic equations by factorization:

Ans.

⇒ 5x(7x−5)=23(2x²−2x−4)
⇒ 35x²−25x=46x²−46x−92
⇒ 46x²−35x²−46x+25x−92 = 0
⇒ 11x²−21x−92 = 0
⇒ 11x²−44x+23x−92 = 0
⇒ 11x(x−4)+23(x−4) = 0
⇒ (11x+23)(x−4) = 0
⇒ 11x+23=0 or x−4 = 0
⇒ x = −23/11 or x = 4

Hence, the factors are 4 and -23/11.