Question

the sum of the squares of two consecutive odd positive integers is 394.Find them

Answer 1

Given that,

the sum of squares of two odd consecutive positive integer is 394.

let the first odd no. be x

therefore the second consecutive no. is = x + 2 (since we know that there's an even no. after every odd no. but here we want two no.s so we'll add 2 with x)

now their sum of squares = 394

⇒ (x)² + (x + 2)² = 394

using identity (a + b)² = a² + 2ab + b²

⇒ x² + x² + 4x + 4 = 394

⇒ 2x² + 4x - 390 = 0

taking 2 as common,

⇒ x² + 2x - 195 = 0

⇒ x² + (15x - 13x) - 195 = 0

⇒ x² + 15x - 13x - 195 = 0

⇒ x(x + 15) - 13(x + 15) = 0

⇒ (x + 15) (x -13)

⇒ x = -15 0r x = 13

since we've to find positive odd no. so we'll eliminate -15.

hence the two consecutive integers are :-

• x = 13

• x + 2 = 13 + 2 = 15

VERIFICATION :-

LHS -

= (13)² + (15)²

= 169 + 225

= 394

RHS -

= 394

∴ LHS = RHS. hence verified!