FINDING TWO NUMBERS WITH THE GIVEN SUM AND PRODUCT

Key Concept

Assume the two numbers as x and y.

Say, the sum of the two numbers x and y is is s.

x + y = s

y = s - x ----(1)

Say, the product of the two numbers x and y is p.

xy = p ----(2)

Solve (1) and (2) for x and y by substitution.

Example 1 :

The sum of two numbers is 60 and their product is 576. Find the numbers.

Solution :

Let x and y be the two numbers.

Given : The sum of two numbers is 60.

 x + y = 60

y = 60 - x ----(1)

Given : The product of two numbers is 576.

xy = 576

Substitute y = 60 - x into the above equation.

x(60 - x) = 576

60x - x2 = 576

-x2 + 60x - 576 = 0

Multiply both sides by -1.

x2 - 60x + 576 = 0

Solve by factoring.

x2 - 12x - 48x + 576 = 0

x(x - 12) - 48(x - 12) = 0

(x - 12)(x - 48) = 0

x - 12 = 0  or  x - 48 = 0

x = 12  or  x = 48

When x = 12,

(1)----> y = 60 - 12

y = 48

When x = 48,

(1)----> y = 60 - 48

y = 12

x = 12  and  y = 48

or

x = 48  and  y = 12

Therefore, the two numbers are 12 and 48.

Verification :

The sum of two numbers is 60.

12 + 48 = 60

60 = 60

The product of the two numbers is 576.

12  48 = 576

576 = 576

The answer is justied.

Example 2 :

The sum of two numbers is ¹⁰⁄₃ and their product is 1. Find the numbers.

Solution :

Let x and y be the two numbers.

Given : The sum of two numbers is ¹⁰⁄₃.

 x + y = ¹⁰⁄₃

y = ¹⁰⁄₃ - x ----(1)

Given : The product of two numbers is 1.

xy = 1

Substitute y = ¹⁰⁄₃ - x into the above equation.

x(¹⁰⁄₃ - x) = 1

(¹⁰⁄₃)x - x2 = 1

Multiply both sides by 3.

3[(¹⁰⁄₃)x - x2] = 3(1)

10x - 3x2 = 3

-3x2 + 10x - 3 = 0

Multiply both sides by -1.

3x2 - 10x + 3 = 0

Solve by factoring.

3x2 - x - 9x + 3 = 0

x(3x - 1) - 3(3x - 1) = 0

(3x - 1)(x - 3) = 0

3x - 1 = 0  or  x - 3 = 0

x =   or  x = 3

When x = ,

(1)----> y = ¹⁰⁄₃ -

y = ⁽¹⁰ ⁻ ¹⁾⁄₃

y = ⁹⁄₃

y = 3

When x = 3,

(1)----> y = ¹⁰⁄₃ - 3

y = ⁽¹⁰ ⁻ ⁹⁾⁄₃

y = 

y = 

x = ⅓  and  y = 3

or

x = 3  and  y =

Therefore, the two numbers are ⅓ and 3.

Verification :

The sum of two numbers is ¹⁰⁄₃.

3 +  = ¹⁰⁄₃

3 +  = ¹⁰⁄₃

⁽⁹ ⁺ ³⁾⁄₃ = ¹⁰⁄₃

¹⁰⁄₃ = ¹⁰⁄₃

The product of the two numbers is 1.

  = 1

1 = 1

The answer is justied.

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