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The sum of the squares of two consecutive odd numbers are 394. Find the product of two
numbers.

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Hint:- The odd consecutive numbers has a difference of two.

\[ \Rightarrow \]Let two consecutive odd numbers be \[x\] and \[x + 2\]
So, according to the given condition in question
\[ \Rightarrow {x^2} + {\left( {x + 2} \right)^2} = 394\] (1)
Now, for solving equation 1. We get,
\[
   \Rightarrow {x^2} + {x^2} + 4x + 4 = 394 \\
   \Rightarrow 2{x^2} + 4x + 4 = 394 \\
\]
Solving above equation. We get,
\[ \Rightarrow {x^2} + 2x - 195 = 0\]
Now, for solving above equation we split 2x, it becomes
\[ \Rightarrow {x^2} + 15x - 13x - 195 = 0\]
Taking common factors of the above equation to get the value of \[x\].
\[
   \Rightarrow x\left( {x + 15} \right) - 13\left( {x + 15} \right) = 0 \\
   \Rightarrow \left( {x + 15} \right)\left( {x - 13} \right) = 0 \\
\]
Hence possible values of \[x\] are -15 and 13.
But x cannot be negative.
\[ \Rightarrow So,{\text{ }}x = 13\]
\[ \Rightarrow \]So, the first odd number will be 13.
\[ \Rightarrow \]And, another odd number will be \[13 + 2 = 15\].
So, we have two find the product of these two odd numbers.
\[ \Rightarrow \]Required product\[ = 13*15 = 195\].
Hence, the product of two consecutive numbers whose sum of squares is 394 is 195.

Note:- Whenever we come up with this type of problem then first assume two numbers such that they depend on each other (to make calculations easy) like here we assume x+2 which depends on x. And then find the value of the variable using the given condition. Then we should multiply two numbers to get the required answer.
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