NCERT Class 10 Mathematics

76 MATHEMATICS Example 6 : Find the dimensions of the prayer hall discussed in Section 4.1. Solution : In Section 4.1, we found ...

QUADRATIC EQUATIONS 77 Therefore, (x – 2)(x + 4) = 2x + 1 i.e., x^2 + 2x – 8 = 2x + 1 i.e., x^2 – 9 = 0 So, Sunita’s present age ...

78 MATHEMATICS The process is as follows: x^2 + 4x =(x^2 + 4 2 x) + 4 2 x =x^2 + 2x + 2x =(x + 2) x + 2 × x =(x + 2) x + 2 × x + ...

QUADRATIC EQUATIONS 79 So, 9x^2 – 15x + 6 = 0 can be written as 512 3 24 ✄ x✂ ✁☎ ✂ ✆ ✝ =0 i.e., 5 2 3 2 ✞✡ x✠ ✟☛ ☞ ✌ = 1 4 So, t ...

80 MATHEMATICS = 5222 5 6336 ☎x✂ ✁✆ ✄ ✂ ✝ ✞ = 5122 5 1^2 636 6 6 ☞✟xx✡ ✠✌ ✡ ☛✟☞ ✡ ✠✌ ✡✟☞ ✠✌ ✍ ✎ ✍ ✎ ✍ ✎ So, the solutions of 3x^ ...

QUADRATIC EQUATIONS 81 Therefore, the solutions of the equations are 3 2 x and 1. Let us verify our solutions. Putting 3 2 x in ...

82 MATHEMATICS Example 9 : Find the roots of 4x^2 + 3x + 5 = 0 by the method of completing the square. Solution : Note that 4x^2 ...

QUADRATIC EQUATIONS 83 If b^2 – 4ac 0, then by taking the square roots in (1), we get 2 x b a ✁ = (^24) 2 bac a ✂ ✄ Therefore, ...

84 MATHEMATICS Example 11 : Find two consecutive odd positive integers, sum of whose squares is 290. Solution : Let the smaller ...

QUADRATIC EQUATIONS 85 x = 325 2 = 35 2 ✁ = 4 or – 1 But x (^) ✂ – 1 (Why?). Therefore, x = 4. So, the breadth of the park = 4m ...

86 MATHEMATICS Example 14 : Find the roots of the following equations: (i) 1 xx3, 0 x ✁ ✂ (ii) 11 3, 0, 2 2 x x x ✄ ✁ ✂ ✄ Solut ...

QUADRATIC EQUATIONS 87 Example 15 : A motor boat whose speed is 18 km/h in still water takes 1 hour more to go 24 km upstream th ...

88 MATHEMATICS Find the roots of the following equations: (i) 1 xx3, 0 x ✁ ✂ (ii) 1111 xx 4730 ✁ ✄ , x ☎ – 4, 7 The sum of ...

QUADRATIC EQUATIONS 89 If b^2 – 4ac = 0, then x =^0 2 b a ✁ , i.e., or – 22 x bb aa ✂ ✄ So, the roots of the equation ax^2 + bx ...

90 MATHEMATICS Now, AB = 13m, and since AB is a diameter, APB = 90° (Why?) Therefore, AP^2 + PB^2 =AB^2 (By Pythagoras theorem) ...

QUADRATIC EQUATIONS 91 EXERCISE 4.4 Find the nature of the roots of the following quadratic equations. If the real roots exist, ...

92 MATHEMATICS A NOTE TO THE READER In case of word problems, the obtained solutions should always be verified with the conditio ...

5 5.1 Introduction You must have observed that in nature, many things follow a certain pattern, such as the petals of a sunflowe ...

94 MATHEMATICS (iv)The number of unit squares in squares with side 1, 2, 3,... units (see Fig. 5.2) are, respectively 12 , 2^2 , ...

ARITHMETIC PROGRESSIONS 95 In the examples above, we observe some patterns. In some, we find that the succeeding terms are obtai ...