Question

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

Answer 1

`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`

Answer 2

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