A=mx2+m2x+2x+2m if n is the only root for that what is the amount of m+n

We are given the equation \( A = mx^2 + m^2x + 2x + 2m \), and we're told that \( n \) is the only root of this equation. We need to find the value of \( m + n \).

### Step 1: Setting up the equation
Since \( n \) is the only root of the equation, we substitute \( x = n \) into the equation and set it equal to zero because roots of a polynomial are the values of \( x \) that make the equation equal to zero.

Thus, we have:
\[
A = mn^2 + m^2n + 2n + 2m = 0
\]

### Step 2: Analyzing the polynomial
To determine the relationship between \( m \) and \( n \), we consider the fact that \( n \) must satisfy the equation above. There may be multiple ways to proceed, including solving for \( m \) and \( n \) simultaneously.

Let me calculate a possible form of \( m + n \) by trying to express \( n \) in terms of \( m \) and simplifying the relationship.

Let me do this calculation.The solutions for \( m + n \) are:

1. \( m + n = m - \frac{2}{m} \)
2. \( m + n = 0 \)

Therefore, depending on the values of \( m \), \( m + n \) could either be \( m - \frac{2}{m} \) or simply 0.

Would you like to explore this further, or do you have any other specific questions?

Here are 5 related questions to think about:
1. What happens if \( m = 2 \) or \( m = -2 \)?
2. How would the situation change if there were more roots?
3. What do the roots tell us about the graph of the polynomial?
4. Can you factor the original polynomial to better understand its properties?
5. How would this scenario extend to higher degree polynomials?

**Tip**: When dealing with quadratic equations, always check if factoring or using the quadratic formula can simplify finding roots.

A = mx^2 + m^2x + 2x + 2m if n is the only root for that what is the amount of m+n?

This guide explains how to solve the polynomial equation A = mx^2 + m^2x + 2x + 2m where n is the only root. Learn key algebraic concepts, the relationship between m and n, and explore the possible values of m + n. Suitable for Grades 9-12.

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