Question

The sum of the squares of two consecutive positive even integers is 52.Find the integers

Answer 1

4 and 6

Explanation:

one number = x

other number = x + 2

x² + (x + 2)² = 52

x² + x² + 4x + 4 = 52

2x² + 4x - 48 = 0

x² + 2x - 24 = 0

(x - 4)(x + 6) = 0

x = 4 or x = -6

Since x is positive

x = 4

x + 2 = 6

Answer 2

The two consecutive positive integers are
`4` and
`6`.

Explanation:

Let
`x` be an even integer, then the next even integer is
`(x+2)`.

The sum of the squares of these even integers is
`52`.

We can write the following equation and solve for
`x`.

`x^2+(x+2)^2=52`.

We expand to get,

`x^2+x^2+4x+4=52`.

We simplify to get,

`2x^2+4x+4-52=0`.

This gives us,

`2x^2+4x-48=0`

We divide through by 2 to get,

`x^2+2x-24=0`.

This is now a quadratic trinomial

We split the middle term to get,

`x^2+6x-4x-24=0`.

We factor to obtain,

`x(x+6)-4(x+6)=0`

`\Rightarrow (x+6)(x-4)=0`

`\Rightarrow (x+6)=0 or (x-4)=0`

`\Rightarrow x=-6 or x=4`

We discard
`x=-6` since it is not a positive integer.

`\therefore x=4`

Hence the two consecutive positive even integers are,

`4`

and

`4+2=6`