Question

The product of two even integers is \[728\]. What are the numbers?

Answer 1

Hint:The integers which are exactly divisible by \[2\] are called the even numbers, while the integers which are not divisible by \[2\] are called odd numbers. Try to solve it using Quadratic factorization using the splitting of the middle term method. In this method the middle term is split into two factors and then the common factors are taken in brackets and this is how we get the factors.

Complete step by step answer:
According to the given question, first let’s take out the consecutive numbers. So, let the even consecutive numbers be \[x\]and \[x\, + \,2\] because from the question we get that both the numbers are even numbers.
It is given that the product of two numbers is \[728\] . So,
\[x(x\, + \,2)\, = \,728\]
\[ \Rightarrow {x^2}\, + \,2x\, = \,728\]
\[ \Rightarrow {x^2}\, + \,2x\, - 728\, = \,0\]
Now, by applying middle term factorisation method,
\[{x^2}\, + \,28x\, - 26x\, - \,728\, = \,0\]
\[ \Rightarrow x(x\, + \,28)\, - \,26(x\, + \,28)\, = \,0\]
\[ \Rightarrow (x\, - \,26)\,(x\, + \,28)\, = \,0\]
\[ \therefore x\, = \,26\,,\, - 28\]

So, the numbers are \[26,\, - 28\].

Additional information:
Consecutive generally means a sequence which cannot be broken or these are integers that follow a sequence in which each subsequent number is one added to the previous number. Factorisation is a method in which a mathematical expression or a polynomial is divided into different terms so that it becomes easier to solve the expression.

Note: Middle term splitting method is easy to find factors, but there is another method that is taking the highest common factor directly and obtaining the factors. This method is working for some questions only, mostly for 3 variable questions. This method makes the question solving very quick and is easy to use.