Question

the sum of the squares of two consecutive even natural number is 164.find the Numbers

Answer 1

Let the first number be
$\bold{ {x}^{2}.....eq.1 }$

Let the second consecutive number be
$\bold {{x}^{2} + 2....eq.2}$
{Even number... so add 2}

According to the given condition,

$\bold{{x}^{2} + ({x}^{2} + 2) = 164 }\\ \bold {2x {}^{2} + 2 = 164} \\ \bold {2x {}^{2} = 164 - 2 }\\ \ \bold{{2x}^{2} = 162}$
$\bold{x {}^{2} = \frac{162}{2 }} \\ \bold{x {}^{2} = 81} \\ \bold{x = \sqrt{81}}$
$\bold{x = 9}$

Hence,From eq.1 and eq.2,
$\bold{{x}^{2} = {9}^{2} = 81} \\ \bold { {x}^{2} + 2 = {9}^{2} + 2 = 81 + 2 =} \\ \bold{ 83}$

Answer 2

Let the first number be = (x)

∴ Second number = (x + 2)

Sum = 164

So, (x)² + (x + 2)² = 164

⇒ x² + x² + 4x + 4 = 164

⇒ 2x² + 4x = 160

⇒ x² + 2x = 80

⇒ x² + 2x - 80 = 0

⇒ (x - 8)(x + 10) = 0

So, x = 8 and x = - 10.

Since (- 10) is not a natural number, so we will take x = 8.

∴ The numbers are:-

8 and 10.