Question

the sum of squares of two consecutive natural numbers is 244; find the number. complete answer

Answer 1

✰ AnSwer :

• The sum of squares of two consecutive natural number is 244, and the numbers are 12 and 10. 

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☯ GiVen :

• Sum of squares of two consecutive natural number is 244.

☯ To Find :

• Original number = ?

☯ SoluTion :

• Let us consider the first number as x, and the second number as x + 2  

According to the question,

$\longmapsto \sf \: ({x})^{2}+({x}+2)^{2}= 244 \\ \\ \longmapsto \sf\: {x}^{2}+{x}^{2}+4 {x}+4= 244 \\ \\ \longmapsto \sf \: 2 x^{2}+4 x=244-4 \\ \\ \longmapsto \sf\: 2 x^{2}+4 x=240 \\ \\ \longmapsto \sf \: 2 x^{2}+4 x-240=0 \\ \\ \longmapsto \sf \: 2\left({x}^{2}+2 {x}-120\right)=0 \\ \\ \longmapsto \sf \: \left({x}^{2}+2 {x}-120 \right)=0 \\ \\ \longmapsto \sf\: {x}^{2}+12 {x}-10 {x}-120 \\ \\ \longmapsto \sf\: {x}({x}+12)-10({x}+12)=0 \\ \\ \longmapsto \sf \: (x-10)(x+12)=0 \\ \\ \longmapsto \:{ \underline{ \boxed{ \bf{ \purple{ \: x=10,-12 \: }}}}} \: \bigstar$

Therefore, as number can't be negative, original number is 10 and +12.

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Answer 2

Solution :-

Let's :-

• First number = x

• 2nd number = x + 2

According to Question :

→ ( x )² + ( x + 2 )² = 244

→ x² + x² + 4 + 4x = 244

→ 2x² + 4x + 4 - 244 = 0

→ 2x² + 4x - 240 = 0

Both side divided by 2

→ x² + 2x - 120 = 0

By splitting middle term

→ x² + 12x - 10x - 120 = 0

→ x ( x + 12 ) - 10 ( x + 12 ) = 0

→ ( x - 10 ) ( x + 12 ) = 0

→ x = 10

→ x = - 12

First number = 10

Second number = 10 + 2 = 12

Verification :-

( 10 )² + ( 12 )² = 244

100 + 144 = 244

244 = 244

LHS = RHS