Question

find the two consecutive positive integers whose sum of their squares is 365​

Answer 1

Answer :

• The consecutive positive integers are 13 and 14.

Given :

• Sum of the squares of two consecutive positive integers

To Find :

• Those two consecutive positive integers.

Solution :

Let

• First consecutive number = x
• Second consecutive number = (x + 1)

Their squares

• First consecutive number = x²
• Second consecutive number = (x + 1)²

It is given that

• Sum of their squares is 365.

According to question :

→ x² + (x + 1)² = 365

→ x² + x² + 2x + 1 = 365

→ 2x² + 2x + 1 = 365

→ 2x² + 2x + 1 - 365 = 0

→ 2x² + 2x - 364 = 0 [ It's in the form of quadratic equation ]

→ x² + x - 182 = 0

→ x² + 14x - 13x - 182 = 0

→ x (x + 14) - 13 (x + 14) = 0

→ (x - 13) (x + 14) = 0

______________

→ x - 13 = 0

→ x = 13

→ x + 14 = 0

→ x = -14

______________

Take the value [ x = 13 ]

Now

• First consecutive positive integer = x = 13
• Second consecutive positive integer = x + 1 = 13 + 1 = 14

Hence, the two consecutive positive integers are 13 and 14.

Answer 2

Given

• Sum of the square of two consecutive positive integers is 365.

⠀

To find

• Two consecutive integers.

⠀

Solution

• Let the

⠀⠀⠀⠀❍ First consecutive integer be x

⠀⠀⠀⠀❍ Second integer be (x + 1)

⠀

★ Now, according to the question

$\boxed{\sf{\orange{x^2 + (x + 1)^2 = 365}}}$

⠀

Identity used

$\large{\boxed{\boxed{\sf{(a + b)^2 = a^2 + b^2 + 2ab}}}}$

⠀

• Let's solve it

⠀

$\tt\longmapsto{x^2 + x^2 + (1)^2 + 2(x)(1) = 365}$

⠀

$\tt\longmapsto{2x^2 + 1 + 2x = 365}$

⠀

$\tt\longmapsto{2x^2 + 2x = 365 - 1}$

⠀

$\tt\longmapsto{2(x^2 + x) = 364}$

⠀

$\tt\longmapsto{x^2 + x = \dfrac{364}{2}}$

⠀

$\tt\longmapsto{x^2 + x = 182}$

⠀

★ Factorising by elimination method

$\tt\longmapsto{x^2 + x - 182 = 0}$

⠀

$\tt\longmapsto{x^2 + 14x - 13x - 182 = 0}$

⠀

$\tt\longmapsto{x(x + 14) - 13(x + 14) = 0}$

⠀

$\tt\longmapsto{(x + 14) (x - 13) = 0}$

⠀

We have,

• x = -14 or 13

⠀

⠀⠀❍ As it is given in the question that ⠀⠀⠀the integer is positive,

⠀

Therefore, we take x = 13

⠀

Hence,

• The two positive consecutive integers are 13 and 14.

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