Question

The product of two consecutive integers is 156. Find the integers.
A) 10 and 13
B) 12 and 13
C) 12 and 11
D) 1 and 13

Answer 1

Hint:
Here, we have to find the two consecutive integers. We will assume the consecutive integers in the form of variables. We will then use the given conditions to frame a quadratic equation. Then we will factorize the equation using the method of factorization and solve it further to find the consecutive integers.

Complete Step by step Solution:
Let \[x\] and \[\left( {x + 1} \right)\] be two consecutive integers respectively.
We are given that the product of two consecutive integers is 156.
So, we get
\[\left( x \right)\left( {x + 1} \right) = 156\]
By multiplying the terms, we get
\[ \Rightarrow \left( {{x^2} + x} \right) = 156\]
By rewriting the equation, we get
\[ \Rightarrow {x^2} + x - 156 = 0\]
Now, we will solve the above quadratic equation by Factorization method.
\[ \Rightarrow {x^2} - 12x + 13x - 156 = 0\]
Now, we will group the terms by common factors, we get
\[ \Rightarrow x\left( {x - 12} \right) + 13\left( {x - 12} \right) = 0\]
\[ \Rightarrow \left( {x - 12} \right)\left( {x + 13} \right) = 0\]
By the zero product property, we get
\[\begin{array}{l} \Rightarrow \left( {x - 12} \right) = 0\\ \Rightarrow x = 12\end{array}\]
Or
\[\begin{array}{l} \Rightarrow \left( {x + 13} \right) = 0\\ \Rightarrow x = - 13\end{array}\]
\[x = - 13\] is not in the multiple choices given.
So, the first number is 12 and the second number is \[12 + 1 = 13\]

Therefore, the two consecutive integers are 12 and 13. Thus Option(B) is the correct answer.

Note:
Here, while factoring the equation we need to follow the rules for the factorization method. The product of the coefficient of \[{x^2}\] and the constant term is the product of the roots. The coefficient of \[x\] is the sum of the roots. The sum of roots is to be factorized according to the product of roots. Then by taking the common factors, we will find the solutions of the quadratic equation. Zero product property states that when the product of two factors is zero, then one of the factors is separately zero i.e., if \[ab = 0\] then \[a = 0\] or \[b = 0\] . The product of either two positive or two negative integers will be positive.