Question

8. The sum of the squares of two consecutive positive even numbers is 52. Find the numbers.​

Answer 1

Let's solve the problem step by step:

Let's assume the first even number as
`\displaystyle\sf x`.

The next consecutive even number can be represented as
`\displaystyle\sf x+2`.

According to the given information, the sum of the squares of these two numbers is 52:

`\displaystyle\sf x^(2) +( x+2)^(2) =52`

Expanding the equation:

`\displaystyle\sf x^(2) +x^(2) +4x+4=52`

Combining like terms:

`\displaystyle\sf 2x^(2) +4x+4=52`

Subtracting 52 from both sides:

`\displaystyle\sf 2x^(2) +4x-48=0`

Dividing the equation by 2 to simplify:

`\displaystyle\sf x^(2) +2x-24=0`

Factoring the quadratic equation:

`\displaystyle\sf ( x+6)( x-4)=0`

Setting each factor equal to zero and solving for
`\displaystyle\sf x`:

`\displaystyle\sf x+6=0` or
`\displaystyle\sf x-4=0`

If
`\displaystyle\sf x+6=0`, then
`\displaystyle\sf x=-6`. However, since we are looking for positive even numbers,
`\displaystyle\sf x=-6` is not valid.

If
`\displaystyle\sf x-4=0`, then
`\displaystyle\sf x=4`.

Therefore, the first even number is 4, and the next consecutive even number is
`\displaystyle\sf 4+2=6`.

So the two numbers are 4 and 6.

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