Question

The product of two consecutive integers is $ 12 $ . What is the second integer?
A. $ 4 $ or $ - 1 $
B. $ 4 $ or $ - 3 $
C. $ 3 $ or $ - 4 $
D. $ 2 $ or $ - 3 $

Answer 1

Hint: Here we will use the concept of consecutive numbers which is defined as the numbers with the difference of $ 1 $ between two consecutive numbers. Here we will assume any variable as the consecutive numbers and then will simplify the expression for the required value.

Complete step-by-step answer:
Let us assume that the first consecutive number to be
And let us assume that the second consecutive number to be
Given that the product of the two consecutive numbers is $ 12 $
Frame the word statement in the form of the mathematical expression –
 $ x(x + 1) = 12 $
Simplify the above expression by multiplying the term outside the bracket with the terms inside the bracket.
 $ {x^2} + x = 12 $
Move all the terms on one side of the equation. When we move any term from one side to the other the sign of the term also changes. Positive term becomes negative and vice-versa.
 $ {x^2} + x - 12 = 0 $
Split the middle term in such a way that the product of its terms is equal to the product of the first and the last term.
 $ {x^2} + 4x - 3x - 12 = 0 $
Make the pair of first two terms and the last two terms.
 $ \underline {{x^2} + 4x} - \underline {3x - 12} = 0 $
Take common multiple common from the paired terms
 $ x(x + 4) - 3(x + 4) = 0 $
Take common multiple common from the above expression
 $ (x + 4)(x - 3) = 0 $
 $
 x + 4 = 0 \\
   \Rightarrow x = - 4 \\
 $
 or
$
x - 3 = 0 \\
   \Rightarrow x = 3 \;
 $
So, the correct answer is “Option C”.

Note: Be careful about the sign convention while splitting the middle term and always consider the sign of the first and the last term. Always frame the correct mathematical expression and solve it wisely. Know the concept of the consecutive numbers and apply accordingly. Consecutive numbers are the numbers back-to-back in the series.