Question

The sum of squares of two consecutive natural numbers is $ 85 $ . Find the numbers.

Answer 1

Hint: As we know that the above given question is a word problem. A problem is a mathematical question written as one sentence or more describing a real life scenario where that problem needs to be solved by the way of mathematical calculation. Also we know that the numbers that follow each other continuously in the order from the smallest to the largest are called consecutive numbers. As for example: $ 1,2,3,4... $ and so on are consecutive numbers.

Complete step-by-step answer:
We need to first understand the requirement of the question which is the sum of squares of two consecutive natural numbers.
Let us assume the two consecutive natural numbers are $ x $ and $ x + 1 $ . We will square both the numbers according to the question i.e. $ {x^2} $ and $ {(x + 1)^2} $ .
By putting them according to the question we have: $ {x^2} + {(x + 1)^2} = 85 $ .
We will solve it now $ \Rightarrow {x^2} + {x^2} + 1 + 2x = 85 $ .
It further gives us the value, $ 2{x^2} + 2x - 84 = 0 $ . Taking the common factor out of all the three : $ {x^2} + x - 42 = 0 $ .
Here we get the expression of a quadratic equation, so will factorise by splitting the middle term by breaking the constant term to give the same results.
So we can say: $ {x^2} + 7x - 6x - 42 = 0 $ . Take the common factors out i.e. $ x(x + 7) - 6(x + 7) = 0 \Rightarrow (x + 7)(x - 6) = 0 $ .
So, either $ x + 7 = 0 $ i.e. $ x = - 7 $ and $ x - 6 = 0 $ i.e. $ x = 6 $ .
We will consider the positive number, so if $ x = 6 $ then, $ x + 1 = 6 + 1 $ i.e. $ 7 $ .
Hence two consecutive natural numbers are $ 6 $ and $ 7 $ .
So, the correct answer is “ $ 6 $ and $ 7 $ ”.

Note: We should note that the formula that we used above in breaking the squares of the numbers is sum of squares formula i.e. $ {(a + b)^2} = {a^2} + {b^2} + 2ab $ . Based on the requirement and by observing all the necessary information that is already available in the question we gather the information and then create an equation or by unitary method whichever is applicable, then we solve the problem and then verify the answer by putting the value in the problem and see whether we get the same answer or not.