Chapter 6 Quadratic Equation Class 10 ICSE Mathematics

Exercise 6

1. Solve the following equation:
(x - 8) (x + 6) = 0
Answer
(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
Answer
(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
Answer
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
Answer
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
Answer
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
Answer
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



Answer

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
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8.
Solve the following:


3x² + 25x + 42 = 0



Answer

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



Answer

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



Answer

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



Answer

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



Answer

√2x² - 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



Answer

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



Answer

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



Answer

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