Question

The product of two consecutive natural numbers is 72. Find the numbers.

Hint: For solving this question we will first assume the two consecutive numbers to be $n,\left( n+1 \right)$ and then write their product in terms of the assumed variable. Then, we will get an equation as per the given data and then solve the equation to find the correct answer.

Given:
Two consecutive natural numbers whose product is 72.
Let, two consecutive natural numbers are $n$ and $\left( n+1 \right)$ .
Now, it is given that the product of the numbers is 72. Then,
$n\times \left( n+1 \right)=72$
Now, we have to solve the above equation to find the value of $n$ .
\begin{align} & n\times \left( n+1 \right)=72 \\ & {{n}^{2}}+n-72=0..........\left( 1 \right) \\ \end{align}
Above equation is a quadratic equation. Now, we will solve this quadratic equation using splitting the middle term for which we have to find two numbers such that their product is -72 and sum is 1. Then, we will split the middle term of the quadratic equation to solve it.
From (1) we have,
\begin{align} & {{n}^{2}}+n-72=0 \\ & \Rightarrow {{n}^{2}}+9n-8n-72=0 \\ & \Rightarrow n\left( n+9 \right)-8\left( n+9 \right)=0 \\ & \Rightarrow \left( n+9 \right)\left( n-8 \right)=0 \\ & \Rightarrow n=-9,8 \\ \end{align}
We have solved the equation and got two values of $n$ that are -9 and 8. But we will consider only the positive values for our solution because it is given that $n$ is a natural number, so it can not be negative and it should be greater than 1.
Now, one of the required natural numbers is $n=8$ . Then, according to our assumption, another number is $n+1=9$ .
Thus, the required two consecutive natural numbers are 8 and 9.

Note: The problem was very easy to solve but one should know how to solve quadratic equations using splitting the middle term technique. Moreover, the student should be careful while splitting the middle term and proceed correctly.