FINDING TWO NUMBERS WITH THE GIVEN SUM AND PRODUCT

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

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 = ⅓  or  x = 3

x = ⅓  and  y = 3

or

x = 3  and  y = ⅓

Therefore, the two numbers are ⅓ and 3.

Verification :

The sum of two numbers is ¹⁰⁄₃.

3 + ⅓ = ¹⁰⁄₃

3 + ⅓ = ¹⁰⁄₃

⁽⁹ ⁺ ³⁾⁄₃ = ¹⁰⁄₃

¹⁰⁄₃ = ¹⁰⁄₃

The product of the two numbers is 1.

3 ⋅ ⅓ = 1

1 = 1

The answer is justied.

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