The sum of two numbers is 18, and the product of these two numbers is 56. What are the numbers?

To solve this problem, we can use the given information about the sum and product of the two numbers. Let's denote the numbers as \( x \) and \( y \).

The problem provides us with two equations:
1. The sum of the two numbers: \( x + y = 18 \)
2. The product of the two numbers: \( xy = 56 \)

We can solve these equations using the quadratic formula. 

First, rewrite the sum equation as \( y = 18 - x \). Substitute \( y \) in the product equation:
\[ x(18 - x) = 56 \]

This simplifies to:
\[ 18x - x^2 = 56 \]

Rearrange the equation into standard quadratic form:
\[ x^2 - 18x + 56 = 0 \]

To find the roots of the quadratic equation, use the quadratic formula:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]

Here, \( a = 1 \), \( b = -18 \), and \( c = 56 \). Plugging in these values:
\[ x = \frac{-(-18) \pm \sqrt{(-18)^2 - 4 \cdot 1 \cdot 56}}{2 \cdot 1} \]
\[ x = \frac{18 \pm \sqrt{324 - 224}}{2} \]
\[ x = \frac{18 \pm \sqrt{100}}{2} \]
\[ x = \frac{18 \pm 10}{2} \]

This gives us two solutions for \( x \):
\[ x = \frac{18 + 10}{2} = \frac{28}{2} = 14 \]
\[ x = \frac{18 - 10}{2} = \frac{8}{2} = 4 \]

Thus, the two numbers are \( x = 14 \) and \( y = 4 \).

Verify:
- Sum: \( 14 + 4 = 18 \)
- Product: \( 14 \times 4 = 56 \)

Both conditions are satisfied, so the numbers are \( 14 \) and \( 4 \).

Would you like more details or have any questions?

Here are 8 related questions for further practice:
1. What are the solutions to the quadratic equation \( x^2 - 10x + 24 = 0 \)?
2. If the sum of two numbers is 20 and their product is 96, what are the numbers?
3. Solve for \( x \) and \( y \) if \( x + y = 12 \) and \( xy = 35 \).
4. Find the roots of the equation \( x^2 - 7x + 10 = 0 \).
5. Given that the sum of two numbers is 22 and their product is 120, determine the numbers.
6. If two numbers add up to 15 and their product is 54, what are the numbers?
7. Solve \( x^2 - 15x + 56 = 0 \) for \( x \).
8. What are the solutions to the quadratic equation \( x^2 + 6x + 9 = 0 \)?

**Tip:** When solving quadratic equations, always check if the quadratic can be factored 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 18 and their product is 56. This problem involves algebraic equations and quadratic formula. Verify the solutions and explore related practice questions.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation