Question

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

Answer 1

Answer:

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

Step-by-step 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

$4+2=6$

Answer 2

Answer:

4 and 6

Step-by-step 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