An integer is 10 less than 4 times another. If the product of the two integers is 50, then find the integers.

Question

An integer is  less than  times another. If the product of the two integers is , then find the integers.

Answer 1

Let X = unknown integer

 

      Y = unknown integer

 

Two unknowns, therefore need two independent equations to solve

 

from problem statement, the two independent equations:

 

   1)  Y = 4*X - 10 

 

   2)  X*Y  = 50

 

Substitute eqn 1) into eqn 2) and solve for X:

 

X*(4*X - 10) = 50        (set up as standard quadratic equation and solve for X as:      a*x^2 +  b*x + c = 0       

 

4*X^2 - 10*X - 50 = 0      

 

divide by 2 all terms to simplify,

 

 2*X^2 - 5*X - 25 = 0                        a = 2     b = -5     c = -25         x = [-b +/- √(b^2 - 4*a*c)]/2*a

 

 

Therefore,  solve for positive root:

 

X  =  [5 + √(25 + 4*2*25)]/4   =  (5 + 15)/4 = 5 

 

Y  =  4*X - 10 = 20 - 10 = 10

 

Checking, X*Y =5*10 = 50

 

Therefore, integers are 5 and 10