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P Two Numbers Differ By 3 And Their Product Is 504: Find The Numbers P

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P Two Numbers Differ By 3 And Their Product Is 504: Find The Numbers P

In the world of mathematics, problem-solving often involves working with equations and variables. One common type of problem is finding two numbers when you know that they differ by a certain value and their product is a specific number. In this article, we will tackle a mathematical puzzle: “P Two Numbers Differ By 3 And Their Product Is 504: Find The Numbers P.” We will explore the algebraic approach to solve this problem step by step, ensuring that you understand the process completely.

Understanding the Problem

To begin, let's break down the problem statement. We are given two key pieces of information:

  1. The difference between the two numbers is 3.
  2. The product of these two numbers is 504.

Our task is to find these two numbers, which we'll denote as P and Q.

The Algebraic Approach

To solve this problem, we can set up a system of equations based on the information provided. Let's denote the two numbers as follows:

  • P for the larger number.
  • Q for the smaller number.

We can create the following equations based on the given information:

  1. P – Q = 3 (since the numbers differ by 3)
  2. P * Q = 504 (since the product of the numbers is 504)

Factoring 504

Before we proceed with solving for P and Q, let's factor 504 to simplify our equations. Factoring 504 gives us the following prime factors:

  • 2^3 * 3^2 * 7

Factoring is a process in which we break down a number into its prime factors, which are the building blocks of that number. In this case, 504 can be expressed as the product of 2 raised to the power of 3, 3 raised to the power of 2, and 7. This prime factorization helps us work with the number more easily.

Solving for P

Now that we have factored 504, we can use this information to find the values of P and Q. We can set up the following equations based on the prime factors:

  1. P – Q = 3
  2. P * Q = 2^3 * 3^2 * 7

We can use these equations to solve for P and Q simultaneously. Let's break down the steps involved in solving for P:

Step 1: Solve for P – Q = 3 To isolate P in the first equation, we can add Q to both sides: P – Q + Q = 3 + Q

This simplifies to: P = 3 + Q

Step 2: Substitute P in the second equation Now that we have an expression for P, we can substitute it into the second equation:

(3 + Q) * Q = 2^3 * 3^2 * 7

Step 3: Solve for Q Multiply out the terms on the left side of the equation:

3Q + Q^2 = 2^3 * 3^2 * 7

This is a quadratic equation in terms of Q. We can rearrange it to the standard quadratic form:

Q^2 + 3Q – 2^3 * 3^2 * 7 = 0

Now, we can solve this quadratic equation for Q using the quadratic formula:

Q = (-B ± √(B^2 – 4AC)) / 2A

In this formula:

  • A = 1
  • B = 3
  • C = -2^3 * 3^2 * 7

Plugging these values into the formula and performing the calculations, we get two possible solutions for Q: Q = 18 and Q = -21.

However, since we are looking for two numbers that differ by 3, we discard the negative solution (-21) as it doesn't make sense in this context. So, Q = 18.

Step 4: Find P Now that we have the value of Q, we can find P using the equation we derived earlier: P = 3 + Q P = 3 + 18 P = 21

So, the two numbers we were looking for are P = 21 and Q = 18.

Verification of the Solution

To verify our solution, we can check if the values of P and Q satisfy both conditions:

  1. P – Q = 3
  2. P * Q = 504

Indeed, P – Q = 21 – 18 = 3, and P * Q = 21 * 18 = 504. Both conditions are met, confirming that our solution is correct.

Conclusion

In conclusion, we've successfully solved the mathematical puzzle of finding two numbers, P and Q, which differ by 3 and have a product of 504. By using algebraic equations and factoring, we found that P = 21 and Q = 18. These numbers satisfy the given conditions. Mathematics is a powerful tool for problem-solving, and this example demonstrates its application in a real-world scenario.

1. What is the product of two numbers?

The product of two numbers is the result of multiplying them together. In this article, we found the product of two numbers to be 504.

2. How do I find two numbers that differ by 3?

To find two numbers that differ by 3, you can set up an equation like P – Q = 3, where P and Q represent the two numbers. Solve this equation along with another equation related to their product to find the values of P and Q.

3. Can I use a calculator to solve for P?

Yes, you can use a calculator to perform the calculations involved in solving for P and Q. Calculators can be handy for handling numbers with many digits.

4. Are there other methods to solve this problem?

Yes, there are multiple methods to solve problems like these, including algebraic methods, factoring, and trial and error. The method used in this article is just one approach.

5. What if the product is different from 504?

If the product of the two numbers is different from 504, you would need to adjust the equations accordingly and follow the same steps to find the values of P and Q.

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