# Frank Solutions for Chapter 6 Quadratic Equation Class 10 ICSE Mathematics

### Exercise 6

1. Solve the following equation:

(x – 8) (x + 6) = 0

(x – 8) (x + 6) = 0

Equate both to zero,

(x – 8) = 0, (x + 6) = 0

⇒ x = 8 or x = -6

2. Solve the following equation:

(2x + 3) (3x – 7) = 0

(2x + 3) (3x – 7) = 0

Equate both to zero,

(2x + 3) = 0, (3x – 7) = 0

⇒ 2x = -3, 3x = 7

⇒ x = -3/2 or x = 7/3

3. Solve the following equation:

4x2 + 16x = 0

4x2 + 16x = 0

Take out common in each terms,

4x(x + 4) = 0

Equate both to zero,

4x = 0, x + 4 = 0

⇒ x = 0, x = – 4

4. Solve the following equation:

2x2 – 3x – 9 = 0

2x2 – 3x – 9 = 0

Divided by 2 for both side of each term we get,

(2x2/2) – (3x/2) – (9/2) = 0/2

⇒ x2 – 3x/2 – 9/2 = 0

⇒ x2 – 3x + (3/2)x – 9/2 = 0

Take out common in each terms,

x(x – 3) + (3/2) (x – 3) = 0

⇒ (x + 3/2) (x – 3) = 0

Equate both to zero,

x + 3/2 = 0, x – 3 = 0

⇒ x = -3/2, x = 3

5. Solve the following equation:

2x2 – x – 6 = 0

2x2 – x – 6 = 0

Divided by 2 for both side of each term we get,

2x2/2 – x/2 – 6/2 = 0

⇒ x2 – x/2 – 3 = 0

⇒ x2 – 2x + (3/2)x – 3 = 0

Take out common in each terms,

x(x – 2) +3/2 (x – 2) = 0

⇒ (x – 2) + (x + (3/2)) = 0

Equate both to zero,

x – 2 = 0, x + 3/2 = 0

⇒ x = 2, x = – 3/2

6. Solve the following question:

5x2 – 11x + 2 = 0

5x2 – 11x + 2 = 0

⇒ 5x2 – 10x – x + 2 = 0

Take out common in each terms,

5x(x – 2) – 1(x – 2) = 0

⇒ (5x – 1) (x – 2) = 0

Equate both to zero,

5x – 1 = 0, x – 2 = 0

⇒ 5x = 1, x = 2

⇒ x = 1/5, x = 2

7. Solve the following equation:

4x2 – 13x – 12 = 0

4x2 -13x – 12 = 0

Divided by 4 for both side of each term we get,

4x2/4 – 13x/4 – 12/4 = 0/4

⇒ x2 – 13x/4 – 3 = 0

⇒ x2 – 4x + 3x/4 – 3 = 0

Take out common in each terms,

x(x – 4) + ¾(x – 4) = 0

⇒ (x – 4) (x + ¾) = 0

Equate both to zero,

(x – 4) = 0, x + ¾ = 0

⇒ x = 4, x = -¾

8. Solve the following:

3x2 + 25x + 42 = 0

3x2 + 25x + 42 = 0

Divided by 3 for both side of each term we get,

3x2/3 + 25x/3 + 42/3 = 0/3

⇒ x2 + 25x/3 + 14 = 0

⇒ x2 + 6x + 7x/3 + 14 = 0

Take out common in each terms,

x(x + 6) + 7/3(x + 6) = 0

⇒ (x + 6) (x + 7/3) = 0

Equate both to zero,

x + 6 = 0, x = -7/3

9. Solve the following equation:

25x(x + 1) = -4

25x(x + 1) = -4

25x2 + 25x = –4

Divided by 25 for both side of each term we get,

25x2/25 + 25x/25 = – 4/25

⇒ x2 + x = – 4/25

⇒ x2 + x + 4/25 = 0

⇒ x2 + (1/5)x + (4/5)x + 4/25 = 0

Take out common in each terms,

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

⇒ (x + 4/5) (x + 1/5) = 0

Equate both to zero,

x + 4/5 = 0, x + 1/5 = 0

⇒ x = -4/5, x = -1/5

10. Solve the following equation:

10x – 1/x = 3

10x – 1/x = 3

⇒ (10x2 – 1)/x = 3

By cross multiplication we get,

10x2 – 1 = 3x

⇒ 10x2 – 3x -1 = 0

Divided by 10 for both side of each term we get,

10x2/10 – 3x/10 – 1/10 = 0/10

⇒ x2 – 3x/10 – 1/10 = 0

⇒ x2 – 1x/5 = ½x – 1/10 = 0

Take out common in each terms,

x(x + 1/5) – ½(x + 1/5) = 0

⇒ (x – ½) (x + 1/5) = 0

Equate both to zero,

x – ½ = 0, x + 1/5 = 0

⇒ x = ½, x = -1/5

11. Solve the following equation:

2/x2 – 5/x + 2 = 0

2/x2 – 5/x + 2 = 0

Multiply by x2 for both side of each term we get,

2x2/x2 – 5x2/x + 2x2 = 0

⇒ 2 – 5x + 2x2 = 0

Above equation can be written as,

2x2 – 5x + 2 = 0

Divided by 2 for both side of each term we get,

2x2/2 – 5x/2 + 2/2 = 0/2

⇒ x2 – 5x/2 + 1 = 0

⇒ x2 – 2x – ½x + 1 = 0

Take out common in each terms,

x(x – 2) – ½(x – 2) = 0

⇒ (x – ½) (x – 2) = 0

Equate both to zero,

x – ½ = 0, x – 2 = 0

⇒ x = ½, x = 2

12. Solve the following equation:

√2x2 – 3x – 2√2 = 0

√2x– 3x – 2√2 = 0

Divided by √2 for both side of each term we get,

√2x2/√2 – 3x/√2 – 2√2/√2 = 0

⇒ x2 – 3x/√2 – 2 = 0

⇒ x2 +(1/√2)x – 2√2x – 2 = 0

Take out common in each terms,

x(x + 1/√2) – 2√2(x + 1/√2) = 0

⇒ (x + 1/√2) (x – 2√2) = 0

Equate both to zero,

x + 1/√2 = 0, x – 2√2 = 0

⇒ x = -1/√2, x = 2√2

13. Solve the following equation:

a2x2 – 3abx + 2b2 = 0

a2x2 – 3abx + 2b2 = 0

Divided by a2 for both side of each term we get,

a2x2/a2 – 3abx/a2 + 2b2/a2 = 0

⇒ x2 – 3bx/a + 2(b/a)2 = 0

⇒ x2 – (b/a)x – 2(b/a)x + 2(b/a)2 = 0

Take out common in each terms,

x(x – b/a) – 2(b/a) (x – b/a) = 0

⇒ (x – 2(b/a)) (x – b/a) = 0

Equate both to zero,

x – 2(b/a) = 0, x – b/a = 0

⇒ x = 2(b/a), x = b/a

14. Solve the following equation:

x2 – (√2 + 1)x + √2 = 0

x2 – (√2 + 1)x + √2 = 0

⇒ x2 – x – √2x + √2 = 0

Take out common in each terms,

x(x – 1) – √2(x – 1) = 0

⇒ (x – 1) (x – √2) = 0

Equate both to zero,

x – 1 = 0, x – √2 = 0

⇒ x = 1, x = √2

15. Solve the following equation:

x2 – (√3 + 1)x + √3 = 0

x2 – (√3 + 1)x + √3 = 0

⇒ x2 – √3x – x + √3 = 0

Take out common in each terms,

x(x – √3) – 1(x – √3) = 0

⇒ (x – √3) (x – 1) = 0

Equate both to zero,

x – √3 = 0, x – 1 = 0

⇒ x = √3, x = 1