The sum of two numbers is 37. Their product is 312. What are the numbers?

2 Answers
Aug 6, 2018

#x = 13, y = 24 and x = 24, y = 13#

Explanation:

Let the numbers be represented by #x and y#

The sum of two numbers is #37#

#x + y = 37#

Their product is #312#

#x xx y = 312#

#xy = 312#

Solving simultaneously;

#x + y = 37 - - - eqn1#

#xy = 312 - - - eqn2#

From #eqn2#

#xy = 312#

Making #x# the subject formula;

#(xy)/y = 312/y#

#(xcancely)/cancely = 312/y#

#x = 312/y - - - eqn3#

Substitute #eqn3# into #eqn1#

#x + y = 37#

#(312/y) + y = 37#

Multiply through by #y#

#y(312/y) + y(y) = y(37)#

#cancely(312/cancely) + y^2 = 37y#

#312 + y^2 = 37y#

#y^2 - 37y + 312 = 0#

Solving the Quadratic Equation..

#y^2 - 37y + 312 = 0#

Using Factorization Method

The factors are, #-13 and -24#

# - 37y = -13y - 24y#

#312 = -13 xx - 24#

Therefore;

#y^2 - 13y - 24y + 312 = 0#

By Grouping;

#(y^2 - 13y) (- 24y + 312) = 0#

Factorizing;

#y(y - 13) -24 (y - 13) = 0#

#(y - 13) (y - 24) = 0#

#y - 13 = 0 or y - 24 = 0#

#y = 13 or y = 24#

Substituting the values of #y# into #eqn3#

#x = 312/y#

When, #y = 13#

#x = 312/13#

#x = 24#

Similarly when, #y = 24#

#x = 312/24#

#x = 13#

Hence;

#x = 13, y = 24 and x = 24, y = 13#

Aug 6, 2018

The two numbers are : 13 and 24

Explanation:

Let #x and y , (x < y )# be the two numbers,such that

sum =#x+y=37=>y=37-xto (1)#

and product # x*y=312...to(2)#

Subst. #y=37-x # into #(2)#

#:.x(37-x)=312#

#:.37x-x^2=312#

#:.x^2-37x+312=0#

Now ,

#(-24)+(-13)=-37 and (-24)xx(-13)=312#

#:.x^2-24x-13x+312=0#

#:.x(x-24)-13(x-24)=0#

#:.(x-24)(x-13)=0#

#:.x-24=0 or x-13=0#

#:.x=24# #or x=13#

So, from #(1)#

#y=13 or y=24#

Hence the two numbers are : 13 and 24