Book:- ML aggarwal, Avichal publication, class10th, quadratic equation in one variable, chapter 5, exercise 5.5

This is an important ques and asked in previous year question paper

(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.

Question no.2 , ML Aggarwal, chapter 5, exercise 5.5, quadratic equation in one variable, ICSE board,

Solution:(i)ÂIf the product of two positive consecutive even integers is 288, find the integers.Let us consider first positive even integer number be â€˜2xâ€™

Second even integer number be â€˜2x + 2â€™

So according to the question,

2x Ã— (2x + 2) = 288

4x

^{2}Â + 4x â€“ 288 = 0Divide by 4, we get

x

^{2}Â + x â€“ 72 = 0Let us factorize,

x

^{2}Â + 9x â€“ 8x â€“ 72 = 0x(x + 9) â€“ 8(x + 9) = 0

(x + 9) (x â€“ 8) = 0

So,

(x + 9) = 0 or (x â€“ 8) = 0

x = -9 or x = 8

âˆ´ Value of x = 8 [since, -9 is not positive]

First even integer = 2x = 2 (8) = 16

Second even integer = 2x + 2 = 2(8) + 2 = 18

(ii)ÂIf the product of two consecutive even integers is 224, find the integers.Let us consider first positive even integer number be â€˜2xâ€™

Second even integer number be â€˜2x + 2â€™

So according to the question,

2x Ã— (2x + 2) = 224

4x

^{2}Â + 4x â€“ 224 = 0Divide by 4, we get

x

^{2}Â + x â€“ 56 = 0Let us factorize,

x

^{2}Â + 8x â€“ 7x â€“ 56 = 0x(x + 8) â€“ 7(x + 8) = 0

(x + 8) (x â€“ 7) = 0

So,

(x + 8) = 0 or (x â€“ 7) = 0

x = -8 or x = 7

âˆ´ Value of x = 7 [since, -8 is not positive]

First even integer = 2x = 2 (7) = 14

Second even integer = 2x + 2 = 2(7) + 2 = 16

(iii)ÂFind two consecutive even natural numbers such that the sum of their squares is 340.Let us consider first positive even natural number be â€˜2xâ€™

Second even number be â€˜2x + 2â€™

So according to the question,

(2x)

^{2}Â + (2x + 2)^{2}Â = 3404x

^{2}Â + 4x^{2}Â + 8x + 4 â€“ 340 = 08x

^{2}Â + 8x â€“ 336 = 0Divide by 8, we get

x

^{2}Â + x â€“ 42 = 0Let us factorize,

x

^{2}Â + 7x â€“ 6x â€“ 56 = 0x(x + 7) â€“ 6(x + 7) = 0

(x + 7) (x â€“ 6) = 0

So,

(x + 7) = 0 or (x â€“ 6) = 0

x = -7 or x = 6

âˆ´ Value of x = 6 [since, -7 is not positive]

First even natural number = 2x = 2 (6) = 12

Second even natural number = 2x + 2 = 2(6) + 2 = 14

(iv)ÂFind two consecutive odd integers such that the sum of their squares is 394.Let us consider first odd integer number be â€˜2x + 1â€™

Second odd integer number be â€˜2x + 3â€™

So according to the question,

(2x + 1)

^{2}Â + (2x + 3)^{2}Â = 3944x

^{2}Â + 4x + 1 + 4x^{2}Â + 12x + 9 â€“ 394 = 08x

^{2}Â + 16x â€“ 384 = 0Divide by 8, we get

x

^{2}Â + 2x â€“ 48 = 0Let us factorize,

x

^{2}Â + 8x â€“ 6x â€“ 48 = 0x(x + 8) â€“ 6(x + 8) = 0

(x + 8) (x â€“ 6) = 0

So,

(x + 8) = 0 or (x â€“ 6) = 0

x = -8 or x = 6

When x = -8, then

First odd integer = 2x + 1 = 2 (-8) + 1 = -16 + 1 = -15

Second odd integer = 2x + 3 = 2(-8) + 3 = -16 + 3 = -13

When x = 6, then

First odd integer = 2x + 1 = 2 (6) + 1 = 12 + 1 = 13

Second odd integer = 2x + 3 = 2(6) + 3 = 12 + 3 = 15

âˆ´ The required odd integers are -15, -13, 13, 15.