Product of two consecutive even number is 120 find the number

Let's solve the problem step by step.

Given:
- The product of two consecutive even numbers is 120.

Let the two consecutive even numbers be \( x \) and \( x + 2 \).

The product of these two numbers is:
\[ x(x + 2) = 120 \]

We can simplify this to:
\[ x^2 + 2x = 120 \]

Rearrange it to form a standard quadratic equation:
\[ x^2 + 2x - 120 = 0 \]

Next, we solve this quadratic equation using the quadratic formula:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]

Here, \( a = 1 \), \( b = 2 \), and \( c = -120 \). Substituting these values in, we get:
\[ x = \frac{-2 \pm \sqrt{2^2 - 4 \cdot 1 \cdot (-120)}}{2 \cdot 1} \]
\[ x = \frac{-2 \pm \sqrt{4 + 480}}{2} \]
\[ x = \frac{-2 \pm \sqrt{484}}{2} \]
\[ x = \frac{-2 \pm 22}{2} \]

This gives us two possible solutions for \( x \):
\[ x = \frac{-2 + 22}{2} = \frac{20}{2} = 10 \]
\[ x = \frac{-2 - 22}{2} = \frac{-24}{2} = -12 \]

Since we are looking for even numbers, we have two pairs of solutions:
1. \( x = 10 \) and \( x + 2 = 12 \)
2. \( x = -12 \) and \( x + 2 = -10 \)

So, the two consecutive even numbers whose product is 120 are \( 10 \) and \( 12 \) (or \(-12\) and \(-10\)).

Would you like further details or have any questions?

Here are 8 relative questions to further your understanding:
1. How do you set up a quadratic equation for consecutive numbers?
2. Why is it important to rearrange the equation into standard form?
3. What does each part of the quadratic formula represent?
4. Can you explain the steps to simplify the quadratic formula?
5. What are the possible values for the product of two numbers?
6. How do you check your solutions in a quadratic problem?
7. What is the significance of the discriminant in the quadratic formula?
8. How can you verify that the numbers found are even?

**Tip:** When solving word problems involving quadratic equations, carefully define your variables and write down all steps to ensure clarity and accuracy.

Learn how to solve the problem where the product of two consecutive even numbers is 120. Understand the quadratic equation involved and its solution using the quadratic formula. Suitable for students in Grades 7-9.

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