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The sum of the squares of 2 consecutive odd numbers is 394. Find the numbers.

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Hint: Consider the 2 consecutive odd numbers as x and x+2. Find the sum of squares of these numbers.Find the value of x and you will get the two consecutive odd numbers.

“Complete step-by-step answer:”
The sum of the squares of 2 consecutive odd numbers is 394. Let us consider one odd number as x and the other consecutive odd number as (x + 2).
We know the odd numbers 1, 3, 5, 7……
So if one number is ‘x’ then the other consecutive odd number can be found by adding 2 to the \[{{1}^{st}}\]number.
So let us take 2 consecutive odd numbers as x and x + 2.
Now it is given that the sum of squares of these consecutive numbers x and (x + 2) is 394.
\[\therefore {{\left( x \right)}^{2}}+{{\left( x+2 \right)}^{2}}=394\]
We know, \[{{\left( a+b \right)}^{2}}={{a}^{2}}+2ab+{{b}^{2}}\].
Now open the brackets and simplify them,
\[\begin{align}
  & {{x}^{2}}+{{x}^{2}}+2\times 2x+{{2}^{2}}=394 \\
 & \Rightarrow 2{{x}^{2}}+4x+4=394 \\
\end{align}\]
Divide the entire equation by 2.
\[\begin{align}
  & {{x}^{2}}+2x+2=197 \\
 & {{x}^{2}}+2x=197-2 \\
 & {{x}^{2}}+2x=195 \\
 & {{x}^{2}}+2x-195=0-(1) \\
\end{align}\]
We got a quadratic equation which is similar to the general quadratic equation, \[a{{x}^{2}}+bx+c=0\].
By comparing equation (1) and the general equation, we get
a = 1, b = 2, c = -195.
Apply these values in the quadratic formula \[\dfrac{-b\pm \sqrt{{{b}^{2}}-4ac}}{2a}\] and find the value of x.
\[\begin{align}
  & \dfrac{-2\pm \sqrt{{{\left( 2 \right)}^{2}}-4\times 1\times \left( -195 \right)}}{2\times 1}=\dfrac{-2\pm \sqrt{4+780}}{2} \\
 & =\dfrac{-2\pm \sqrt{784}}{2}=\dfrac{-2\pm \sqrt{28\times 28}}{2}=\dfrac{-2\pm 28}{2} \\
\end{align}\]
Hence the roots are \[\left( \dfrac{-2+28}{2} \right)\]and \[\left( \dfrac{-2-28}{2} \right)\]= 13 and -15.
\[\therefore \]Value of x = 13, which is an odd number.
Thus we got the \[{{1}^{st}}\]consecutive number as x =13.
Hence, \[{{2}^{nd}}\]consecutive number as x + 2 = 13 + 2 = 15
Thus the 2 consecutive odd numbers are 13 and 15.

Note: You should consider 2 consecutive terms as x and (x + 2), which is the key to solve this question. We know an odd number, for example 3 is an odd number. (3 + 2) gives 5, which is the odd number near to 3. Thus, 3 and 5 are consecutive terms.