Question

The sum of the squares of two consecutive positive integers is 265, the integers are

Answer 1

Hint: We will assume the first positive integer to be x and the second positive integer to be x+1. Then we will take the summation of the squares and equate it to 265.

Complete step-by-step answer:
Before proceeding with the question we should understand the concept of integers.
The word integer originated from the Latin word “Integer” which means whole. It is a special set of whole numbers composed of zero, positive numbers and negative numbers and denoted by the letter Z. Examples of Integers – 1, 6, 15. Fractions, decimals, and percents are out of this basket.
Now two consecutive positive integers are mentioned in the question, so let the first positive integer be x and the second positive integer be x+1.
Sum of the squares of these two positive numbers is equal to 265. So forming an equation using this information we get,
\[\Rightarrow {{x}^{2}}+{{(x+1)}^{2}}=265........(1)\]
Now squaring both the terms in left hand side of the equation (1) we get,
\[\Rightarrow {{x}^{2}}+{{x}^{2}}+1+2x=265........(2)\]
Now adding all the like terms in the left hand side of the equation (2) together we get,
\[\Rightarrow 2{{x}^{2}}+2x+1=265........(3)\]
Now rearranging the terms in equation (3) we get,
\[\Rightarrow 2{{x}^{2}}+2x-264=0........(4)\]
Now taking 2 out from all the terms in equation (4) we get,
\[\Rightarrow {{x}^{2}}+x-132=0........(5)\]
Solving the quadratic equation (5) by factoring we get,
\[\begin{align}
  & \Rightarrow {{x}^{2}}-11x+12x-132=0 \\
 & \Rightarrow x(x-11)+12(x-11)=0 \\
 & \Rightarrow (x-11)(x+12)=0.....(6) \\
\end{align}\]
Solving for x in equation (6) we get,
\[\Rightarrow x=-12,\,11\]
Since a positive integer is asked so the two consecutive positive integers are 11 and 12.

Note: Here understanding the concept of integers is the key and also we need to be careful while writing the answer because we may in a hurry write both -12 and 11 as the answer but here only positive integers are asked and hence we need to be careful.