The difference of two natural numbers is 7 and their product is 450. Find the numbers.

Answer

To solve the problem, we need to find two natural numbers based on the conditions given: their difference is 7 and their product is 450.

### Step-by-step Solution:

1. **Define the Variables**:
Let the first number be \( a \) and the second number be \( b \).

2. **Set Up the Equations**:
According to the problem:
- The difference of the two numbers is 7:
\[ a - b = 7 \quad \text{(Equation 1)} \]
- The product of the two numbers is 450:
\[ a \times b = 450 \quad \text{(Equation 2)} \]

3. **Express \( a \) in terms of \( b \)**:
From Equation 1, we can express \( a \) as:
\[ a = b + 7 \quad \text{(Substituting into Equation 1)} \]

4. **Substitute \( a \) in Equation 2**:
Now substitute \( a \) in Equation 2:
\[ (b + 7) \times b = 450 \]
Expanding this gives:
\[ b^2 + 7b = 450 \]

5. **Rearrange into Standard Quadratic Form**:
Rearranging the equation:
\[ b^2 + 7b - 450 = 0 \quad \text{(Equation 3)} \]

6. **Solve the Quadratic Equation**:
We can use the quadratic formula \( b = \frac{-B \pm \sqrt{B^2 - 4AC}}{2A} \), where \( A = 1, B = 7, C = -450 \):
\[ b = \frac{-7 \pm \sqrt{7^2 - 4 \times 1 \times (-450)}}{2 \times 1} \]
Simplifying this:
\[ b = \frac{-7 \pm \sqrt{49 + 1800}}{2} \]
\[ b = \frac{-7 \pm \sqrt{1849}}{2} \]
\[ b = \frac{-7 \pm 43}{2} \]

7. **Calculate Possible Values for \( b \)**:
This gives us two potential solutions:
- \( b = \frac{36}{2} = 18 \)
- \( b = \frac{-50}{2} = -25 \) (not valid since \( b \) must be a natural number)

8. **Find \( a \)**:
Now, substitute \( b = 18 \) back into Equation 1 to find \( a \):
\[ a = b + 7 = 18 + 7 = 25 \]

9. **Conclusion**:
The two natural numbers are:
- First number \( a = 25 \)
- Second number \( b = 18 \)

### Final Answer:
The two natural numbers are 25 and 18.

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