the sum of two numbers is 30. Their product is 176. What are the two numbers?

Let's denote the two numbers as \( x \) and \( y \).

We are given two key pieces of information:

1. The sum of the two numbers: 
   \[
   x + y = 30
   \]

2. The product of the two numbers: 
   \[
   xy = 176
   \]

We can solve this system of equations by first expressing one variable in terms of the other from the sum equation. 

From \( x + y = 30 \), we can express \( y \) as:
\[
y = 30 - x
\]

Now, substitute this expression for \( y \) into the product equation:
\[
x(30 - x) = 176
\]
Expanding and rearranging the equation:
\[
30x - x^2 = 176
\]
\[
x^2 - 30x + 176 = 0
\]

This is a quadratic equation in the standard form \( ax^2 + bx + c = 0 \), where \( a = 1 \), \( b = -30 \), and \( c = 176 \).

We can solve this quadratic equation using the quadratic formula:
\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]
Substituting the values:
\[
x = \frac{-(-30) \pm \sqrt{(-30)^2 - 4(1)(176)}}{2(1)}
\]
\[
x = \frac{30 \pm \sqrt{900 - 704}}{2}
\]
\[
x = \frac{30 \pm \sqrt{196}}{2}
\]
\[
x = \frac{30 \pm 14}{2}
\]

So, the two possible solutions for \( x \) are:
\[
x = \frac{30 + 14}{2} = \frac{44}{2} = 22
\]
or
\[
x = \frac{30 - 14}{2} = \frac{16}{2} = 8
\]

Thus, the two numbers are 22 and 8.

### Verification
- Sum: \( 22 + 8 = 30 \)
- Product: \( 22 \times 8 = 176 \)

Both conditions are satisfied.

### Final Answer:
The two numbers are **22** and **8**.

Would you like more details or have any questions?

Here are five related questions:
1. How can we solve quadratic equations using factoring?
2. What is the discriminant in a quadratic equation, and how is it used?
3. Can you find the roots of the quadratic equation using the completing the square method?
4. How do we derive the quadratic formula from a general quadratic equation?
5. What are the applications of quadratic equations in real-world problems?

**Tip:** Always check your solution by substituting the values back into the original equations to ensure they satisfy all given conditions.

Learn how to solve a mathematical problem where the sum of two numbers is 30 and their product is 176. Understand the steps involved in solving systems of equations and applying the quadratic formula. Suitable for high school students and beyond.

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