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.