Question

Let \(m\) and \(n\) be positive integers, If \(x^2+mx+2n=0\) and \(x^2+2nx+m=0\) have real roots, then the smallest possible value of \(m+n\) is

• A. 7
• B. 8
• C. 5
• D. 6

Answer 1

The Correct Option is D

The problem involves two quadratic equations: \(x^2+mx+2n=0\) and \(x^2+2nx+m=0\). For the quadratics to have real roots, their discriminants must be non-negative. The discriminant \(\Delta\) of a quadratic \(ax^2+bx+c=0\) is given by \(b^2-4ac\).

Step 1: Calculate discriminant of the first equation:

\(\Delta_1 = m^2 - 8n \geq 0\) 

Step 2: Calculate discriminant of the second equation:

\(\Delta_2 = (2n)^2 - 4m = 4n^2 - 4m \geq 0\)

From step 1:

\(m^2 \geq 8n\) (Equation 1)

From step 2:

\(n^2 \geq m\) (Equation 2)

Now, solve these inequalities to find the smallest value of \(m+n\).

Suppose \(m=n\). Plug \(m=n\) into Equation 1:

\(n^2 \geq 8n\)

\(n^2 - 8n \geq 0\)

\(n(n-8) \geq 0\)

Since \(n > 0\), we get \(n \geq 8\). But let's examine smaller integers to fulfill both inequalities.

Try \(m=4\), \(n=2\) as a possible solution:

- Equation 1: \(4^2 \geq 8 \times 2\) results in \(16 \geq 16\), which is true.

- Equation 2: \(2^2 \geq 4\) results in \(4 \geq 4\), which is also true.

Thus, \(m=4\) and \(n=2\) satisfy both conditions with the smallest sum, so \(m+n=6\).

Therefore, the smallest possible value of \(m+n\) is 6.

Answer 2

In order for both quadratic equations to have real roots, the discriminant must be greater than or equal to zero.

Therefore, we must have:

\[ m^2 - 8n \geq 0 \quad \text{and} \quad 4n^2 - 4m \geq 0 \] \[ \Rightarrow m^2 \geq 8n \quad \text{and} \quad n^2 \geq m \]

We're asked to find the minimum value of \( m + n \) for positive integers \( m \) and \( n \) satisfying the above inequalities.

Trying small integer values:

• Let \( n = 2 \Rightarrow n^2 = 4 \Rightarrow m \leq 4 \)
• Now try \( m = 4 \Rightarrow m^2 = 16 \Rightarrow 8n \leq 16 \Rightarrow n \leq 2 \)
• So, \( m = 4 \) and \( n = 2 \) satisfy both conditions.

Therefore, the minimum value of \( m + n \) is:

\[ m + n = 4 + 2 = \boxed{6} \]