**1. Check whether the following are quadratic equations:**

**Solution:**

**Solution:**

**3. In each of the following, determine whether the given numbers are solutions of the given equation or not:**

**Solution:**

**EXERCISE 5.2**

**Solve the following equations (1 to 24) by factorization:**

^{2}– 5x = 0

**2. (i) (x – 3) (2x + 5) = 0**

**(ii) x (2x + 1) = 6**

**Solution:**

^{2}+ x – 6 = 0

^{2}+ 4x – 3x – 6 = 0

**3. (i) x² – 3x – 10 = 0**

**(ii) x(2x + 5) = 3**

**Solution:**

^{2}– 3x – 10 = 0

^{2}– 3x – 10 = 0

^{2}– 5x + 2x – 10 = 0

^{2}+ 5x – 3 = 0

^{2}+ 6x – x – 3 = 0

**4. (i) 3x**

^{2}– 5x – 12 = 0**(ii) 21x**

^{2}– 8x – 4 = 0**Solution:**

^{2}– 5x – 12 = 0

^{2}– 5x – 12 = 0

^{2}– 9x + 4x – 12 = 0

^{2}– 8x – 4 = 0

^{2}– 8x – 4 = 0

^{2}– 14x + 6x – 4 = 0

**5. (i) 3x**

^{2}= x + 4**(ii) x(6x – 1) = 35**

**Solution:**

^{2}= x + 4

^{2}– x – 4 = 0

^{2}– 4x + 3x – 4 = 0

^{2}-1/3x = 1

^{2}– x = 3

^{2}– x – 3 = 0

^{2}– 3x + 2x – 3 = 0

**7. (i) (x – 4)**

^{2}+ 52 = 132**(ii) 3(x – 2)**

^{2}= 147**Solution:**

^{2}+ 52 = 13

^{2}

^{2}– 8x + 16 ++ 25 = 169

^{2}– 8x + 41 – 169 = 0

^{2}– 8x – 128 = 0

^{2}– 16x + 8x – 128 = 0

^{2}= 147

^{2}– 4x + 4) = 147

^{2}– 12x + 12 = 147

^{2}– 12x +12 – 147 = 0

^{2}– 12x – 135 = 0

^{2}– 4x – 45 = 0

^{2}– 9x + 5x – 45 = 0

**8. (i) 1/7(3x – 5)**

^{2}= 28**(ii) 3(y**

^{2}– 6) = y(y + 7) – 3**Solution:**

^{2}= 28

^{2}= 28 × 7

^{2}= 196

^{2}– 30x + 25 = 196

^{2}– 30x + 25 – 196 = 0

^{2}– 30x – 171 = 0

^{2}– 10x – 57 = 0

^{2}– 19x + 9x – 57 = 0

^{2}– 6) = y(y + 7) – 3

^{2}– 18 = y

^{2}+ 7y – 3

^{2}– 18 – y

^{2}– 7y + 3 = 0

^{2}– 7y – 15 = 0

^{2}– 10y + 3y – 15 = 0

**9. x**

^{2}– 4x – 12 = 0, when x ∈ N**Solution:**

^{2}– 4x – 12 = 0

^{2}– 6x + 2x – 12 = 0

**11. 2x**

^{2}– 8x – 24 = 0 when x ∈ I**Solution:**

^{2}– 8x – 24 = 0

^{2}– 4x – 12 = 0

^{2}– 6x + 2x – 12 = 0

**12. 5x**

^{2}– 8x – 4 = 0 when x ∈ Q**Solution:**

^{2}– 8x – 4 = 0

^{2}– 10x + 2x – 4 = 0

**13. 2x**

^{2}– 9x + 10 = 0, when**(i) x ∈ N**

**(ii) x ∈ Q**

**Solution:**

^{2}– 9x + 10 = 0

^{2}– 4x – 5x + 10 = 0

**14. (i) a²x² + 2ax + 1 = 0,a ≠ 0**

**(ii) x² – (p + q)x + pq = 0**

**Solution:**

**15. a²x² + (a² + b²)x + b² = 0,a≠0**

**Solution:**

**EXERCISE 5.3**

**Solve the following (1 to 8) equations by using the formula:**

**1. (i) 2x² – 7x + 6 = 0**

**(ii) 2x² – 6x + 3 = 0**

**Solution:**

**2. (i) x**

^{2}+ 7x – 7 = 0**(ii) (2x + 3) (3x – 2) + 2 = 0**

**Solution:**

^{2}+ 7x – 7 = 0

**3. (i) 256x² – 32x + 1 = 0**

**(ii) 25**

**x²**

**+ 30x + 7 = 0**

**Solution:**

**7. (i) x-1/x = 3,x ≠ 0**

**(ii) 1/ x + 1/(x-2) = 3,x ≠ 0,2**

**Solution:**

^{2}– 1 = 3x

^{2}– 3x – 1 = 0

**10. Solve the following equation by using quadratic equations for x.**

**(i) x² – 5x – 10 = 0**

**(ii) 5x(x + 2) = 3**

**Solution:**

**12. Solve the following equation: x-18/x = 6. Give your answer correct to two x significant figures.**

**Solution:**

^{2}– 18 = 6x

^{2}– 6x – 18 = 0

**EXERCISE 5.4**

**1. Find the discriminate of the following equations and hence find the nature of roots:**

**(i) 3x² – 5x – 2 = 0**

**(ii) 2x² – 3x + 5 = 0**

**(iii) 7x² + 8x + 2 = 0**

**(iv) 3x² + 2x – 1 = 0**

**(v) 16x² – 40x + 25 = 0**

**(vi) 2x² + 15x + 30 = 0**

**Solution:**

**3. Find the nature of the roots of the following quadratic equations:**

**(i) x² – 1/2x – 1/2 = 0**

**(ii) x² – 2√3x – 1 = 0 If real roots exist, find them.**

**Solution:**

^{2 }– 8m + 4 – 4m – 20

^{2}– 12m – 16

^{2}– 12m – 16 = 0

^{2}– 3m – 4 = 0

^{2}– 4m + m – 4 = 0

**7. Find the values of k for which each of the following quadratic equation has equal roots:**

**(i) 9x**

^{2}+ kx + 1 = 0**(ii) x**

^{2}– 2kx + 7k – 12 = 0**Also, find the roots for those values of k in each case.**

**Solution:**

^{2}+ kx + 1 = 0

^{2})– 4ac

^{2 }– 4 (9) (1)

^{2}– 36

^{2}– 36 = 0

^{2}+ kx + 1 = 0

^{2}+ 6x + 1 = 0

^{2}+ 2(3x)(1) + 1

^{2}= 0

^{2}= 0

^{2}+ kx + 1 = 0

^{2}– 6x + 1 = 0

^{2}– 2(3x)(1) + 12 = 0

^{2}– 2kx + 7k – 12 = 0

^{2}– 4ac

^{2}– 4 (1) (7k – 12)

^{2}– 28k + 48

^{2}– 28k + 48 = 0

^{2}– 7k + 12 = 0

^{2}– 3k – 4k + 12 = 0

**8. Find the value(s) of p for which the quadratic equation (2p + 1)x² – (7p + 2)x + (7p – 3) = 0 has equal roots. Also find these roots.**

**Solution:**

^{2}+ 28p – 4p + 16 = 0

**9. If – 5 is a root of the quadratic equation 2x² + px – 15 = 0 and the quadratic equation p(x² + x) + k = 0 has equal roots, find the value of k.**

**Solution:**

**10. Find the value(s) of p for which the equation 2x² + 3x + p = 0 has real roots.**

**Solution:**

**11. Find the least positive value of k for which the equation x² + kx + 4 = 0 has real roots.**

**Solution:**

**Exercise -5.5**

**1. (i) Find two consecutive natural numbers such that the sum of their squares is 61.**

**(ii) Find two consecutive integers such that the sum of their squares is 61.**

**Solution:**

**2. (i) If the product of two positive consecutive even integers is 288, find the integers.**

**(ii) If the product of two consecutive even integers is 224, find the integers.**

**(iii) Find two consecutive even natural numbers such that the sum of their squares is 340.**

**(iv) Find two consecutive odd integers such that the sum of their squares is 394.**

**Solution:**

**8. (i) Find three successive even natural numbers, the sum of whose squares is 308.**

**(ii) Find three consecutive odd integers, the sum of whose squares is 83.**

**Solution:**

**9. In a certain positive fraction, the denominator is greater than the numerator by 3. If 1 is subtracted from both the numerator and denominator, the fraction is decreased by 1/14. Find the fraction.**

**Solution:**

**10. The sum of the numerator and denominator of a certain positive fraction is 8. If 2 is added to both the numerator and denominator, the fraction is increased by 4/35. Find the fraction.**

**Solution:**

**11. A two digit number contains the bigger at ten’s place. The product of the digits is 27 and the difference between two digits is 6. Find the number.**

**Solution:**

**12. A two digit positive number is such that the product of its digits is 6. If 9 is added to the number, the digits interchange their places. Find the number.**

**Solution:**

**13. A rectangle of area 105 cm² has its length equal to x cm. Write down its breadth in terms of x. Given that the perimeter is 44 cm, write down an equation in x and solve it to determine the dimensions of the rectangle.**

**Solution:**

**14. A rectangular garden 10 m by 16 m is to be surrounded by a concrete walk of uniform width. Given that the area of the walk is 120 square meters, assuming the width of the walk to be x, form an equation in x and solve it to find the value of x.**

**Solution:**

**15. (i) Harish made a rectangular garden, with its length 5 meters more than its width. The next year, he increased the length by 3 meters and decreased the width by 2 meters. If the area of the second garden was 119 sqm, was the second garden larger or smaller ?**

**(ii) The length of a rectangle exceeds its breadth by 5 m. If the breadth was doubled and the length reduced by 9 m, the area of the rectangle would have increased by 140 m². Find its dimensions.**

**Solution:**