Question

The product of two successive multiples of 4 is 28 more than the first multiple of 4. Find the multiples.

Answer 1

Hint – In this question the condition on multiples of 4 is given to us. So a multiple of 4 can be written as 4x where x belongs to positive integers and the successive multiple of 4 will be 4x+4 where x belongs to positive integers, so use this concept to get the answer.

“Complete step-by-step answer:”
It is given that the numbers are successive multiples of 4.
So, let the numbers be $4x$ and $\left( {4x + 4} \right)$.
Now it is given that the product of two successive multiples of 4 is 28 more than the first multiple of 4.
So, construct the linear equation according to this information we have,
Product of $4x$ and $\left( {4x + 4} \right)$ is equal to 28 plus $4x$.
$ \Rightarrow 4x\left( {4x + 4} \right) = 28 + 4x$
Now simplify the above equation we have,
$ \Rightarrow 16{x^2} + 16x = 28 + 4x$
$ \Rightarrow 16{x^2} + 12x - 28 = 0$
Now divide by 4 in above equation we have,
$ \Rightarrow 4{x^2} + 3x - 7 = 0$
Now factories the above equation we have,
$ \Rightarrow 4{x^2} + 7x - 4x - 7 = 0$
$
   \Rightarrow x\left( {4x + 7} \right) - 1\left( {4x + 7} \right) = 0 \\
   \Rightarrow \left( {x - 1} \right)\left( {4x + 7} \right) = 0 \\
   \Rightarrow x - 1 = 0,{\text{ }}\left( {4x + 7} \right) = 0 \\
   \Rightarrow x = 1,\dfrac{{ - 7}}{4} \\
 $
Therefore the multiples is
$
  4x = 4, - 7 \\
  4x + 4 = 8, - 3 \\
$
Hence the multiples are (4 and 8) or (-7 and -3).
So this is the required multiples.

Note – Whenever we face such types of problems the key concept is simply to assume the multiple of 4 and then to write the consecutive multiple of 4. Now using the relevant relations provided in the questions and using the information equations can be formulated. Use these equations to get the right answer.