Question

The difference of the squares of two positive consecutive even integers is 84. Find the integers.

Answer 1

Answer:

The two integers are 20 and 22.

Step-by-step explanation:

Let x be the first integer.

$(x+2)^{2}-x^{2} = 84\\x^{2} +4x+4-x^{2} = 84\\4x = 80\\x = 20$

Answer 2

Applying the difference of squares, and solving the equation, it is found that the numbers are 20 and 22.

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• The two consecutive even integers are: n and n + 2.
• The difference of the squares is given by:

$D_s = (n+2)^2 - n^2$

Expanding the difference:

$D_s = n^2 + 4n + 4 - n^2$

$D_s = 4n + 4$

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The difference is 84, thus:

$D_s = 84$

Now, we solve for n.

$4n + 4 = 84$

$4n = 80$

$n = \frac{80}{4}$

$n = 20$

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Thus, the numbers are: $n = 20$ and $n + 2 = 20 + 2 = 22$

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