Question

The sum of the square of two consecutive positive odd integers is 202, how do you find the integers?

Answer 1

Hint: In this problem we have to find two consecutive positive odd integers. We know that the difference between any two positive consecutive odd numbers is 2. Therefore, when we are going to assume any variable as a positive odd number; we have to maintain the difference of 2 for another consecutive odd number.

Formula used:
\[{(a + b)^2} = {a^2} + {b^2} + 2ab\]

Complete step by step answer:
Let us assume the first positive odd number is ‘a’. Then another number will become ‘a+2’.
Now according to question square of first odd number will become
\[ \Rightarrow {a^2}\]
Then square of other number will become
\[ \Rightarrow {(a + 2)^2}\]
And also sum of square of two consecutive positive odd integers will become
\[ \Rightarrow {a^2} + {(a + 2)^2}\]
According to the given problem it will become equal to 202. Therefore,
\[{a^2} + {(a + 2)^2} = 202\]
Now expanding it by using above given formula
\[ \Rightarrow {a^2} + {a^2} + {2^2} + 2 \times a \times 2 = 202\]
Now adding \[{a^2} + {a^2}\]and multiplying \[2 \times a \times 2\]
\[ \Rightarrow 2{a^2} + {2^2} + 4a = 202\]
Now replacing \[{2^2} = 2 \times 2\] we get,
\[ \Rightarrow 2{a^2} + 2 \times 2 + 4a = 202\]
Multiplying \[2 \times 2\] and subtracting 4 from both sides, we get
\[ \Rightarrow 2{a^2} + 4 + 4a - 4 = 202 - 4\]
\[ \Rightarrow 2{a^2} + 4a = 198\]
Taking 198 at left hand side, we get
\[ \Rightarrow 2{a^2} + 4a - 198 = 0\]
Taking 2 common, we get
\[ \Rightarrow 2({a^2} + 2a - 99) = 0\]
Dividing by 2 on both sides, we get
\[ \Rightarrow {a^2} + 2a - 99 = 0\]
Solving above quadratic equation by splitting middle term, we get
\[ \Rightarrow {a^2} + 11a - 9a - 99 = 0\]
Taking common \['a'\] and \[ - 9\],we get
\[ \Rightarrow a(a + 11) - 9(a + 11) = 0\]
Taking common \[(a + 11)\], we get
\[ \Rightarrow (a + 11)(a - 9) = 0\]
Therefore, ‘\[a\]’gives us two value
\[ \Rightarrow a = - 11\]and \[9\]
Assumed number is a positive integer. Therefore,
\[ \Rightarrow a = 9\]
The other number will become
\[ \Rightarrow a + 2\]
\[ \Rightarrow 9 + 2 = 11\]

So, required number are 9 and 11.

Note: We have to take care about what is given in the problem. Either positive or negative integers are given. Sometimes one odd and other even number are given. According to that we have to assume the first number and maintain the difference with the other one.