Question

The difference between any two numbers is 2 and their product is 1443. Find the numbers.

Answer 1

Hint: In the given question, we are given the relation between two numbers thus we have two unknown quantities so we assume the numbers to be any two different variables such that one of the numbers is greater than the other. Then using the given conditions, we can construct two different equations in terms of the two variables and can obtain their values by solving the equations.

Complete step-by-step answer:
Let one of the numbers be x and the other number by y such that $ x > y $ .
We are given that the difference between these two numbers is 2, so –
 $ x - y = 2 $
Rearranging the terms we get, $ x = 2 + y $
The product of the two terms is given as $ xy = 1443 $
Putting the value of x in the above equation, we get a quadratic equation as follows –
 $
  (2 + y)y = 1443 \\
   \Rightarrow {y^2} + 2y - 1443 = 0 \\
   \Rightarrow {y^2} + 39y - 37y - 1443 = 0 \\
   \Rightarrow y(y + 39) - 37(y + 39) = 0 \\
   \Rightarrow (y + 39)(y - 37) = 0 \\
   \Rightarrow y = - 39,y = 37 \;
  $
We know, $ x = 2 + y $ so,
 $
  x = 2 + ( - 39),x = 2 + 37 \\
   \Rightarrow x = - 37,x = 39 \;
  $
Hence the two numbers are either 39 and 37 or -37 and -39.
So, the correct answer is “ 39 and 37 or -37 and -39.”.

Note: On putting the value of x, the equation obtained is the quadratic polynomial equation as the highest power of the variable used in the equation that is called the degree of the polynomial equation is two, the equation is solved by using factorization method in the solution, it can also be solved using completing the square method. Both sets of numbers obtained satisfy the given and assumed conditions so both of them are correct.