Question

Find two consecutive odd positive integers, sum of whose square is 290.

Answer 1

Hint: Our first step to solve this problem is to assume variables for finding the required numbers. Here, we need to take variables for two consecutive odd numbers. For this we will assume two numbers as $ x $ and $ x + 2 $ . After that, by using the given condition, we will obtain a quadratic equation by solving which we can find the required two consecutive odd positive integers.

Complete step-by-step answer:
Let the two consecutive odd positive integers be $ x $ and $ x + 2 $ .
We are given that the sum of squares of these numbers is 290. Therefore, we can say that
 $ {x^2} + {\left( {x + 2} \right)^2} = 290 $
Now, we need to solve this equation to find the value of $ x $ .
We know that $ {\left( {a + b} \right)^2} = {a^2} + 2ab + {b^2} $ . We will now apply this to $ {\left( {x + 2} \right)^2} $
 $ \Rightarrow {\left( {x + 2} \right)^2} = {x^2} + 2x + 4 $
Now, we will put this value in the obtained equation and solve it.
 $
   \Rightarrow {x^2} + {x^2} + 2x + 4 = 290 \\
   \Rightarrow 2{x^2} + 4x - 286 = 0 \\
   \Rightarrow {x^2} + 2x - 143 = 0 \;
  $
We have the quadratic equation. First, we will do its factorization by splitting the middle term.
For splitting the middle term, we need to find two numbers whose sum is 2 and product is -143.
There are two numbers +13 and -11 whose sum is 2 and product is -143.
Therefore, we can write the middle term $ 2x = + 13x - 11x $
Putting this in the quadratic equation,
 $
   \Rightarrow {x^2} + 2x - 143 = 0 \\
   \Rightarrow {x^2} + 13x - 11x - 143 = 0 \\
   \Rightarrow x\left( {x + 13} \right) - 11\left( {x + 13} \right) = 0 \\
   \Rightarrow \left( {x + 13} \right)\left( {x - 11} \right) = 0 \;
  $
Here, we have two possible solutions.
First one is: $ x + 13 = 0 \Rightarrow x = - 13 $
-13 is a negative odd number and we are asked to find two consecutive positive odd numbers. Therefore, -13 is not our answer.
The second solution is: $ x - 11 = 0 \Rightarrow x = 11 $
11 is a positive odd integer. Therefore, our first required integer is 11.
We have taken the second odd number as $ x + 2 $
 $ x + 2 = 11 + 2 = 13 $
Thus, our final answer is: the two consecutive odd positive integers, sum of whose square is 290 are 11 and 13.
So, the correct answer is “11 and 13”.

Note: In this type of question, whenever we are asked to find two consecutive odd or even integers with some condition, then be careful when assuming two numbers. Another thing to keep in mind when solving this type of question is splitting the middle term of the quadratic equation correctly at the time of factorization. This is because a slight mistake in signs can lead us to the incorrect answer.