Question

the sum of squares of two consecutive natural number added to their product is equal to 217 find the numbers​

Answer 1

Question:

The sum of squares of two consecutive natural number added to their product is equal to 217.Find the numbers​?

To find:

The two numbers

Given:

• The sum of squares of two consecutive natural number added to their product is equal to 217.

Solution:

Let,the numbers be $x^2$ and $(x + 1)^2$

$x^2 + (x + 1)^2=217\\\\ \to or,2x^2+1=217\\\\ \to or,2x^2=217-1\\\\\ \to or,2x^2=216\\\\ \to or,x^2=108\\\\\\ \to or,x=\sqrt{2 * 2 * 3 * 3 * 3} \\\\\\ \to or,x=6\sqrt{3}$

Answer:

The two numbers are:$6\sqrt{3}$ and $(6\sqrt{3} + 1)$

$\left \{ {{1st=6\sqrt{3} } \atop {2nd=6\sqrt{3} + 1 }} \right.$

Answer 2

$\large\underline{\sf{Solution-}}$

Let assume that

➢ First consecutive natural number be x

➢ Second consecutive natural number be x + 1

According to statement

The sum of squares of two consecutive natural number added to their product is equal to 217.

So,

$\rm :\longmapsto\: {x}^{2} + {(x + 1)}^{2} + x(x + 1) = 217$

$\rm :\longmapsto\: {x}^{2} + {x}^{2} + 1 + 2x+ {x}^{2} +x = 217$

$\rm :\longmapsto\: 3{x}^{2} + 3x + 1= 217$

$\rm :\longmapsto\: 3{x}^{2} + 3x + 1 - 217 = 0$

$\rm :\longmapsto\: 3{x}^{2} + 3x - 216 = 0$

$\rm :\longmapsto\: 3({x}^{2} + x - 72) = 0$

$\rm :\longmapsto\: {x}^{2} + x - 72 = 0$

$\rm :\longmapsto\: {x}^{2} + 9x - 8x - 72 = 0$

$\rm :\longmapsto\:x(x + 9) - 8(x + 9) = 0$

$\rm :\longmapsto\:(x + 9)(x - 8) = 0$

$\rm\implies \:x = 8 \: \: \: or \: \: \: x = - 9 \: \: \{rejected \}$

$\rm\implies \:\boxed{\tt{ \: \: x \: \: = \: \: 8 \: \: }}$

So,

➢ First consecutive natural number = 8

➢ Second consecutive natural number = 9

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Verification

First consecutive natural number = 8

Second consecutive natural number = 9

According to statement

The sum of squares of two consecutive natural number added to their product is equal to 217.

So,

$\rm :\longmapsto\: {8}^{2} + {9}^{2} + 8 \times 9 = 217$

$\rm :\longmapsto\: 64 + 81 + 72 = 217$

$\rm :\longmapsto\: 217 = 217$

HENCE, VERIFIED