The difference between integers is 5. their product is 500. find the numbers

General
1

the difference between integers is 5. their product is 500. find the numbers.

2

What are two numbers whose difference is 5 and whose product is 500?

Answer:

To solve this problem, we need to set up an equation based on the conditions given: the difference between the two integers is 5, and their product is 500.

Step 1: Setting Up the Equations

  1. Define the Variables:

    • Let the two integers be x and y.
    • Given that x - y = 5.
    • Given that x \cdot y = 500.

  2. Rewrite x in Terms of y:

    • From the equation x - y = 5, we can express x as:
      $$x = y + 5$$

Step 2: Substituting and Solving

  1. Substitute x in the Product Equation:

    • Substitute x = y + 5 into the equation x \cdot y = 500:
      $$(y + 5) \cdot y = 500$$

  2. Expand and Simplify:

    • Expand the equation:
      $$y^2 + 5y = 500$$

  3. Rearrange into Standard Quadratic Form:

    • Rearrange the terms to get a standard quadratic equation:
      $$y^2 + 5y - 500 = 0$$

Step 3: Solving the Quadratic Equation

  1. Use the Quadratic Formula:

    • The quadratic formula for ax^2 + bx + c = 0 is:
      y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
    • Here, a = 1, b = 5, c = -500.

  2. Calculate the Discriminant:

    • Calculate the discriminant (b^2 - 4ac):
      $$5^2 - 4 \cdot 1 \cdot (-500) = 25 + 2000 = 2025$$

  3. Find y Using the Quadratic Formula:

    • Substitute the values into the quadratic formula:
      y = \frac{-5 \pm \sqrt{2025}}{2}
    • Evaluate the square root and solve:
      • \sqrt{2025} = 45

  4. Calculate the Two Possible Values for y:

    • y = \frac{-5 + 45}{2} = 20
    • y = \frac{-5 - 45}{2} = -25

Step 4: Determine Corresponding Values for x

  1. For y = 20:

    • Substitute back to find x:
      $$x = y + 5 = 20 + 5 = 25$$

  2. For y = -25:

    • Substitute back to find x:
      $$x = y + 5 = -25 + 5 = -20$$

Final Answer:

The two pairs of integers that satisfy the given conditions are:

  • (20, 25)
  • (-25, -20)

Both pairs meet the criteria of having a difference of 5 and a product of 500.