Question

An integer is subtracted from its square. The result could be which of the following?
A. A negative even integer
B. An odd integer
C. The product of two consecutive even integers
D. The product of two consecutive odd integers
E. The product of two consecutive integers

Answer 1

Hint: Here we are given a statement and we are asked to find the result of the given statement. To obtain the result, we shall consider an example. We need to deal with the example integer with the statement and we need to analyze the obtained result. Then we can follow the statement in the general format.

Complete step-by-step answer:
The given statement is “An integer is subtracted from its square”. We need to find the result of the given statement.
Before getting into general form, we shall deal with the given statement by an example.
The statement tells us to consider an integer.
We can consider a negative integer or a positive integer.
For an easy-going, we shall consider a positive integer.
Let the integer be $4$.
Then the next step of the given statement is to take the square of the integer.
That is, we need to square $4$.
Thus, we have ${4^2} = 16$
Then according to the statement, we need to subtract the integer from its square.
That is, we have $16 - 4$
Now, we shall take the common number outside.
$16 - 4 = 4\left( {4 - 1} \right)$
$ = 4 \times 3$
We can note that $3$ and $4$are consecutive numbers.
Hence the result is to multiply the two consecutive numbers $3$ and $4$
Now, we shall denote the given statement in general format.
Let the integer be $n$.
Then the next step of the given statement is to take the square of the integer.
That is, we need to square $n$.
Thus, we have ${n^2}$
Then according to the statement, we need to subtract the integer from its square.
That is, we have ${n^2} - n$
Now, we shall take the common number outside.
${n^2} - n = n\left( {n - 1} \right)$
We can note that $n$ and $n - 1$are consecutive numbers.
Hence the result is to multiply the two consecutive numbers $n$ and $n - 1$
Also, when subtract an integer from its square, the result will be the integer (not rational or irrational number)
Therefore, the product of two consecutive integers is the required result, and option E is the correct answer.

So, the correct answer is “Option E”.

Note: The consecutive numbers or consecutive integers are numbers that usually follow each other in sequential order. Generally, the consecutive integers are in ascending or increasing order.
Example, $1,2,3,4,..$
Here the consecutive integer differs by adding $1$ to the previous integer.