Certainly! Let's solve the given problem step by step.
We are given the roots of the quadratic equation \(3x^2 - 2mx + 2n = 0\) as \(x = 2\) and \(x = 3\). We will use these roots to find the values of \(m\) and \(n\).
### Step 1: Understand the Relationship of Roots and Coefficients
For a quadratic equation \(ax^2 + bx + c = 0\), Vieta's formulas tell us the relationships between the roots and the coefficients:
1. The sum of the roots \((p + q) = -\frac{b}{a}\)
2. The product of the roots \((pq) = \frac{c}{a}\)
In the quadratic equation \(3x^2 - 2mx + 2n = 0\), we have:
- \(a = 3\)
- \(b = -2m\)
- \(c = 2n\)
### Step 2: Use the Sum of the Roots
Given roots \(x_1 = 2\) and \(x_2 = 3\):
[tex]\[ x_1 + x_2 = 2 + 3 = 5 \][/tex]
According to Vieta's formulas for the sum of the roots:
[tex]\[ x_1 + x_2 = -\frac{b}{a} \][/tex]
[tex]\[ 5 = -\frac{-2m}{3} \][/tex]
Solving for \(m\):
[tex]\[ 5 = \frac{2m}{3} \][/tex]
[tex]\[ 5 \times 3 = 2m \][/tex]
[tex]\[ 15 = 2m \][/tex]
[tex]\[ m = \frac{15}{2} \][/tex]
[tex]\[ m = 7.5 \][/tex]
### Step 3: Use the Product of the Roots
The product of the roots \(x_1 \cdot x_2 = 2 \cdot 3 = 6\).
According to Vieta's formulas for the product of the roots:
[tex]\[ x_1 x_2 = \frac{c}{a} \][/tex]
[tex]\[ 6 = \frac{2n}{3} \][/tex]
Solving for \(n\):
[tex]\[ 6 \times 3 = 2n \][/tex]
[tex]\[ 18 = 2n \][/tex]
[tex]\[ n = \frac{18}{2} \][/tex]
[tex]\[ n = 9 \][/tex]
### Conclusion
By using Vieta's formulas and the given roots, we found:
- The value of \(m\) is \(7.5\)
- The value of \(n\) is \(9\)
Thus, [tex]\(m = 7.5\)[/tex] and [tex]\(n = 9\)[/tex].
