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

This is an important ques and asked in exam

(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.

Question no.1 , ML Aggarwal, chapter 5, exercise 5.5, quadratic equation in one variable, ICSE board,

Solution:(i)ÂFind two consecutive natural numbers such that the sum of their squares is 61.Let us consider first natural number be â€˜xâ€™

Second natural number be â€˜x + 1â€™

So according to the question,

x

^{2}Â + (x + 1)^{2}Â = 61let us simplify the expression,

x

^{2}Â + x^{2}Â + 1^{2}Â + 2x â€“ 61 = 02x

^{2}Â + 2x â€“ 60 = 0Divide by 2, we get

x

^{2}Â + x â€“ 30 = 0Let us factorize,

x

^{2}Â + 6x â€“ 5x â€“ 30 = 0x(x + 6) â€“ 5 (x + 6) = 0

(x + 6) (x â€“ 5) = 0

So,

(x + 6) = 0 or (x â€“ 5) = 0

x = -6 or x = 5

âˆ´ x = 5 [Since -6 is not a positive number]

Hence the first natural number = 5

Second natural number = 5 + 1 = 6

(ii)ÂFind two consecutive integers such that the sum of their squares is 61.Let us consider first integer number be â€˜xâ€™

Second integer number be â€˜x + 1â€™

So according to the question,

x

^{2}Â + (x + 1)^{2}Â = 61let us simplify the expression,

x

^{2}Â + x^{2}Â + 1^{2}Â + 2x â€“ 61 = 02x

^{2}Â + 2x â€“ 60 = 0Divide by 2, we get

x

^{2}Â + x â€“ 30 = 0Let us factorize,

x

^{2}Â + 6x â€“ 5x â€“ 30 = 0x(x + 6) â€“ 5 (x + 6) = 0

(x + 6) (x â€“ 5) = 0

So,

(x + 6) = 0 or (x â€“ 5) = 0

x = -6 or x = 5

Now,

If x = -6, then

First integer number = -6

Second integer number = -6 + 1 = -5

If x = 5, then

First integer number = 5

Second integer number = 5 + 1 = 6