The sum of two numbers is 11 in the product is 30 find numbers

Given the problem, we need to find two numbers \( x \) and \( y \) such that:

1. Their sum is 11: \( x + y = 11 \)
2. Their product is 30: \( xy = 30 \)

We can solve this problem using a system of equations. 

First, express \( y \) in terms of \( x \) using the first equation:
\[ y = 11 - x \]

Next, substitute \( y \) into the second equation:
\[ x(11 - x) = 30 \]
\[ 11x - x^2 = 30 \]
\[ x^2 - 11x + 30 = 0 \]

This is a quadratic equation. We can solve for \( x \) using the quadratic formula:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
where \( a = 1 \), \( b = -11 \), and \( c = 30 \).

Substitute the values into the formula:
\[ x = \frac{11 \pm \sqrt{(-11)^2 - 4 \cdot 1 \cdot 30}}{2 \cdot 1} \]
\[ x = \frac{11 \pm \sqrt{121 - 120}}{2} \]
\[ x = \frac{11 \pm \sqrt{1}}{2} \]
\[ x = \frac{11 \pm 1}{2} \]

So, we have two solutions for \( x \):
\[ x = \frac{12}{2} = 6 \]
\[ x = \frac{10}{2} = 5 \]

Therefore, the two numbers are \( x = 6 \) and \( y = 5 \), or \( x = 5 \) and \( y = 6 \).

The numbers are \( \boxed{5 \text{ and } 6} \).

Would you like details on this solution or have any questions?

Here are 8 related questions that expand on this information:

1. What are the properties of quadratic equations?
2. How do you derive the quadratic formula?
3. Can a quadratic equation have complex roots? If so, under what conditions?
4. What are the different methods to solve a system of equations?
5. How can you determine if a given quadratic equation can be factored easily?
6. How do you solve a quadratic equation by completing the square?
7. What is the significance of the discriminant in a quadratic equation?
8. How can you verify the solutions of a quadratic equation?

**Tip:** When solving quadratic equations, always check if the equation can be factored easily before using the quadratic formula. This can save time and simplify the process.

Learn how to solve the problem where the sum of two numbers is 11 and their product is 30. This problem involves solving a system of equations and covers key algebraic concepts, including the quadratic formula and system of equations. Suitable for students in Grades 7-9.

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