One number is 5 greater than another. The product of the numbers are 84. Find both the two positive and two negative sets of numbers.

Let x, y be the two numbers.

Given that one number is 5 greater than another.

Let x be the smaller number ans y be the greater number.

That is y=x+5. Let this be the first equation.

And also given that product of the two numbers is 84.

That is x*y = 84, let us plugin y=x+5 here.

x * (x+5) = 84

x^2 + 5x - 84 = 0.

x^2+12x-7x-84 = 0

x (x+12) - 7 (x+12) = 0

(x-7) (x+12) = 0

That is x = 7 or - 12.

If x=7, y = 7+5=12.

If x=-12, y = - 12+5 = - 7.

Hence two positive numbers corresponding to given conditions are 7,12.

And two negative numbers corresponding to given conditions are - 12,-7.

GIVEN

one number is greater than the other by 5.

the product of both are 84.

find out the number.

To proof =

let assume that one number be x

other number be x+5

the equation becomes

⇒ x (x+8) = 84

⇒ x²+5x-84 = 0

⇒ x² + 12x-7x-84 = 0

⇒ (x-7) (x+12) = 0

⇒x=7, x=-12

now the positive numbers are 7 and 12

now the negative numbers are - 12 and - 7

hence proved