Numbers Finder by Difference and Product

Enter the difference and product of two numbers to find them instantly!

Calculate Two Numbers from Their Difference and Product

Result

How to Calculate Two Numbers from Their Difference and Product

Given:

  • \( D \): the difference between the two numbers (\( x > y \))
  • \( P \): the product of the two numbers

Steps:

  1. Express relationships: \( x - y = D, \quad x \cdot y = P\)
  2. Substitute \( x = y + D \): \( (D + y) \cdot y = P\)
  3. Form a quadratic equation: \( y^2 + Dy - P = 0\)
  4. Solve using the quadratic formula: \( y = \frac{-D \pm \sqrt{D^2 + 4P}}{2}\)
  5. Find \( x \): \( x = D + y\)

Examples

Example 1: The difference between two numbers is 20, and their product is 125. Find the numbers.

Solution:

1. Quadratic equation:

\( y^2 + 20y - 125 = 0 \)

2. Discriminant:

\( \sqrt{20^2 + 4 \cdot 125} = \sqrt{900} = 30 \)

3. Roots:

\( y_1 = \frac{-20 + 30}{2} = 5 \)

\( y_2 = \frac{-20 - 30}{2} = -25 \)

4. Calculate \( x \):

\( x = 20 + 5 = 25 \)

\( x = 20 - 25 = -5 \)

Result: The pairs of numbers are ( 25, 5) or (-5, -25).

Example 2: The difference between two numbers is 11, and the product is 1452. Find the numbers.

Solution:

1. Quadratic equation:

\( y^2 + 11y - 1452 = 0 \)

2. Discriminant:

\( \sqrt{11^2 + 4 \cdot 1452} = \sqrt{5929} = 77 \)

3. Roots:

\( y_1 = \frac{-11 + 77}{2} = 33 \)

\( y_2 = \frac{-11 - 77}{2} = -44 \)

4. Calculate \( x \):

\( x_1 = 11 + 33 = 44 \)

\( x_2 = 11 - 44 = -33 \)

Result: The pairs of numbers are (44, 33) or (-33, -44).

Example 3: The difference is 7, and the product is 450. Find the numbers.

Solution:

1. Quadratic equation:

\( y^2 + 7y - 450 = 0 \)

2. Discriminant:

\( \sqrt{7^2 + 4 \cdot 450} = \sqrt{1849} = 43 \)

3. Roots:

\( y_1 = \frac{-7 + 43}{2} = 18 \)

\( y_2 = \frac{-7 - 43}{2} = -25 \)

4. Calculate \( x \):

\( x_1 = 7 + 18 = 25 \)

\( x_2 = 7 - 25 = -18 \)

Result: The pairs of numbers are (25, 18) or (-18, -25).