If `alpha` and `beta` are the roots of the quadratic equation `3m^2+2m-4=0`, then `alpha^(2)+beta^(2)=`

Options

A. `28/9`
B. `(-28)/9`
C. `12/9`
D. `(-12)/9`

Answer

To find the value of \( \alpha^2 + \beta^2 \) for the roots of the quadratic equation \( 3m^2 + 2m - 4 = 0 \), we can use the relationship between the roots and the coefficients of the equation.

### Step-by-Step Solution:

1. **Identify the coefficients**:
The quadratic equation is in the form \( ax^2 + bx + c = 0 \). Here, \( a = 3 \), \( b = 2 \), and \( c = -4 \).

2. **Calculate the sum and product of the roots**:
- The sum of the roots \( \alpha + \beta \) is given by the formula:
\[ \alpha + \beta = -\frac{b}{a} = -\frac{2}{3} \]
- The product of the roots \( \alpha \beta \) is given by the formula:
\[ \alpha \beta = \frac{c}{a} = \frac{-4}{3} \]

3. **Use the identity to find \( \alpha^2 + \beta^2 \)**:
We can use the identity:
\[ \alpha^2 + \beta^2 = (\alpha + \beta)^2 - 2\alpha\beta \]
Substituting the values we calculated:
- \( (\alpha + \beta)^2 = \left(-\frac{2}{3}\right)^2 = \frac{4}{9} \)
- \( 2\alpha\beta = 2 \times \left(\frac{-4}{3}\right) = -\frac{8}{3} \)

4. **Combine the results**:
Now substitute these into the identity:
\[ \alpha^2 + \beta^2 = \frac{4}{9} - \left(-\frac{8}{3}\right) \]
To combine these, convert \( -\frac{8}{3} \) to have a common denominator:
\[ -\frac{8}{3} = -\frac{24}{9} \]
Therefore,
\[ \alpha^2 + \beta^2 = \frac{4}{9} + \frac{24}{9} = \frac{28}{9} \]

5. **Final Result**:
Thus, the value of \( \alpha^2 + \beta^2 \) is:
\[ \alpha^2 + \beta^2 = \frac{28}{9} \]