Find two numbers that multiply to give 200 and have a difference of 35

Let's solve the problem by setting up two equations based on the given conditions.

1. Let the two numbers be \( x \) and \( y \).
2. The product of the numbers is given as:  
   \[
   x \cdot y = 200
   \]
3. The difference between the numbers is given as:  
   \[
   |x - y| = 35
   \]

Since \( x \) and \( y \) are positive numbers, we can assume without loss of generality that \( x > y \). Therefore, the difference becomes:
\[
x - y = 35
\]

Now, we have two equations:
\[
x \cdot y = 200 \quad \text{(1)}
\]
\[
x - y = 35 \quad \text{(2)}
\]

### Step 1: Express \( x \) in terms of \( y \)
From equation (2):
\[
x = y + 35
\]

### Step 2: Substitute into the first equation
Substitute \( x = y + 35 \) into equation (1):
\[
(y + 35) \cdot y = 200
\]
Expanding this:
\[
y^2 + 35y = 200
\]

### Step 3: Solve the quadratic equation
Rearrange the equation to get:
\[
y^2 + 35y - 200 = 0
\]

To solve this quadratic equation, use the quadratic formula:
\[
y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]
where \( a = 1 \), \( b = 35 \), and \( c = -200 \).

Substituting these values:
\[
y = \frac{-35 \pm \sqrt{35^2 + 800}}{2}
\]
\[
y = \frac{-35 \pm \sqrt{1225 + 800}}{2}
\]
\[
y = \frac{-35 \pm \sqrt{2025}}{2}
\]
\[
y = \frac{-35 \pm 45}{2}
\]

### Step 4: Calculate the roots
This gives us two possible values for \( y \):
1. \( y = \frac{-35 + 45}{2} = \frac{10}{2} = 5 \)
2. \( y = \frac{-35 - 45}{2} = \frac{-80}{2} = -40 \) (which is negative and doesn't fit the positive number assumption)

So, \( y = 5 \).

### Step 5: Find \( x \)
Now, substitute \( y = 5 \) back into the equation \( x = y + 35 \):
\[
x = 5 + 35 = 40
\]

### Final Answer:
The two numbers are \( 40 \) and \( 5 \).

Would you like more details, or do you have any questions?

Here are some related questions:
1. How would you solve the problem if the difference was given as 25?
2. What are the steps to solve a quadratic equation by completing the square?
3. Can you find other pairs of numbers that multiply to 200?
4. How does the quadratic formula help in solving such problems?
5. What are the real-world applications of solving such number problems?

**Tip:** When solving quadratic equations, always check for extraneous solutions that might not fit the original problem conditions.

Explore how to find two numbers whose product is 200 and have a difference of 35 through the application of quadratic equations. This detailed guide walks you through solving the quadratic equation step by step, suitable for students in Grades 9-10.

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