Question
Find the consecutive even integers whose squares have the sum $340$.
✓
Answer
Let the consecutive even integers be $2 x$ and $2 x+2$
Then according to the given hypothesis,
$(2 x)^2+(2 x+2)^2=340$
$4 x^2+4 x^2+8 x+4-340=0$
$\Rightarrow 8 x^2+8 x-336=0$
$\Rightarrow x^2+x-42=0$
$\Rightarrow x^2+7 x-6 x-42=0$
$\Rightarrow x(x+7)-6(x+7)=0$
$\Rightarrow(x+7)(x-6)=0$
$\Rightarrow x=-7 \text { or } x=6$
Considering, the positive integers of $x=6$
$\Rightarrow 2 x=12 \text { and } 2 x+2=14$
$\therefore$ The two consecutive even integers are $12$ and $14$
Then according to the given hypothesis,
$(2 x)^2+(2 x+2)^2=340$
$4 x^2+4 x^2+8 x+4-340=0$
$\Rightarrow 8 x^2+8 x-336=0$
$\Rightarrow x^2+x-42=0$
$\Rightarrow x^2+7 x-6 x-42=0$
$\Rightarrow x(x+7)-6(x+7)=0$
$\Rightarrow(x+7)(x-6)=0$
$\Rightarrow x=-7 \text { or } x=6$
Considering, the positive integers of $x=6$
$\Rightarrow 2 x=12 \text { and } 2 x+2=14$
$\therefore$ The two consecutive even integers are $12$ and $14$
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