The product of two whole numbers is 416416 and their sum is 4242. What are the two numbers?




Let's denote the two whole numbers as \( x \) and \( y \). We are given two conditions:

1. The product of the two numbers is \( x \times y = 416416 \).
2. The sum of the two numbers is \( x + y = 4242 \).

We need to find \( x \) and \( y \). This can be done by solving a system of equations:

1. \( x \times y = 416416 \)
2. \( x + y = 4242 \)

### Step 1: Express one variable in terms of the other.
From the sum equation:
\[
y = 4242 - x
\]

### Step 2: Substitute into the product equation.
Substitute \( y = 4242 - x \) into the product equation:
\[
x \times (4242 - x) = 416416
\]

This leads to a quadratic equation:
\[
x(4242 - x) = 416416
\]
\[
4242x - x^2 = 416416
\]
\[
x^2 - 4242x + 416416 = 0
\]

### Step 3: Solve the quadratic equation.
We can solve this quadratic equation using the quadratic formula:
\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]
For the equation \( x^2 - 4242x + 416416 = 0 \), \( a = 1 \), \( b = -4242 \), and \( c = 416416 \).

Let's calculate this.The solutions to the quadratic equation are approximately:

- \( x_1 \approx 4141.45 \)
- \( x_2 \approx 100.55 \)

Since the two numbers are whole numbers, we round them to:

- \( x_1 = 4141 \)
- \( x_2 = 101 \)

Thus, the two whole numbers are **4141** and **101**.

Would you like more details or have any questions?

Here are 5 related questions:
1. What is the difference between these two numbers?
2. What are the prime factors of 4141 and 101?
3. If the sum of two numbers were 5000 and their product was 416416, what would the numbers be?
4. How would you solve this problem graphically?
5. Can this method be used to find two numbers given any sum and product?

**Tip:** The quadratic formula is a powerful tool for solving quadratic equations when factoring isn't easy.

Explore how to solve a problem involving two whole numbers where their product is 416416 and their sum is 42. This math problem is solved using algebraic techniques, specifically focusing on systems of equations and the quadratic formula. Suitable for Grades 7-9 students and anyone interested in algebraic problem-solving.

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