Find two consecutive odd positive integers, sum of whose squares is 29

1.	Find two consecutive odd positive integers, sum of whose squares is 290
Answer» Answer: 11 and 13 Given: The two numbers are consecutive odd positive integers. Sum of their squares is 290 To Find: The two integers. Solution: The integers are odd consecutive, so, there difference should be 2. let the first integer = x The second integer = x + 2 According to the question ⟹ ( x )² + ( x + 2)² = 290 By opening the whole squares, ⟹ x² + x² + 4X + 4 = 290 ⟹ x² + x² + 4x + 4 - 290 = 0 ⟹ 2x² + 4x - 286 = 0 ⟹ 2( x² + 2x - 286) = 0 ⟹ x² + 2x - 286 = 0 By splitting the middle term, ⟹ x² + 13x - 11x - 143 = 0 ⟹ x ( x + 13 ) - 11 ( x + 13 ) = 0 ⟹ ( x + 13 ) ( x - 11 ) = 0 Now, ⟹ (x + 13) = 0 ⟹ x = -13 - 13 is not possible as the integers should be positive OR ⟹ (x - 11 ) = 0 ⟹ x = 11 First integer = x = 11 Second integer = (x + 2) = 13 11 and 13 are the integers. Verification: Sum of their squares should be 290 ⟹ (11)² + (13)² ≟ 290 ⟹ 121 + 169 ≟ 290 ⟹ 290 = 290 LHS = RHS verified !

Find two consecutive odd positive integers, sum of whose squares is 290

Answer»

Answer:

11 and 13

Given:

The two numbers are consecutive odd positive integers.
Sum of their squares is 290

To Find:

The two integers.

Solution:

The integers are odd consecutive, so, there difference should be 2.

let the first integer = x

The second integer = x + 2

According to the question

⟹ ( x )² + ( x + 2)² = 290

By opening the whole squares,

⟹ x² + x² + 4X + 4 = 290

⟹ x² + x² + 4x + 4 - 290 = 0

⟹ 2x² + 4x - 286 = 0

⟹ 2( x² + 2x - 286) = 0

⟹ x² + 2x - 286 = 0

By splitting the middle term,

⟹ x² + 13x - 11x - 143 = 0

⟹ x ( x + 13 ) - 11 ( x + 13 ) = 0

⟹ ( x + 13 ) ( x - 11 ) = 0

Now,

⟹ (x + 13) = 0

⟹ x = -13

- 13 is not possible as the integers should be positive

⟹ (x - 11 ) = 0

⟹ x = 11

First integer = x = 11

Second integer = (x + 2) = 13

11 and 13 are the integers.

Verification:

Sum of their squares should be 290

⟹ (11)² + (13)² ≟ 290

⟹ 121 + 169 ≟ 290

⟹ 290 = 290

LHS = RHS

verified !