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

Options

A. `(1)/(4)`
B. `(7)/(4)`
C. `(-7)/(4)`
D. `(-1)/(4)`

Answer

To solve the problem of finding \( \alpha^2 + \beta^2 \) given the roots \( \alpha \) and \( \beta \) of the quadratic equation \( 2x^2 - 3x + 4 = 0 \), we can follow these steps:

### Step 1: Identify the coefficients
The given quadratic equation is \( 2x^2 - 3x + 4 = 0 \).
Here, we can identify:
- \( a = 2 \)
- \( b = -3 \)
- \( c = 4 \)

### Step 2: Calculate the sum of the roots
The sum of the roots \( \alpha + \beta \) can be calculated using the formula:
\[ \alpha + \beta = -\frac{b}{a} \]
Substituting the values of \( b \) and \( a \):
\[ \alpha + \beta = -\frac{-3}{2} = \frac{3}{2} \]

### Step 3: Calculate the product of the roots
The product of the roots \( \alpha \beta \) can be calculated using the formula:
\[ \alpha \beta = \frac{c}{a} \]
Substituting the values of \( c \) and \( a \):
\[ \alpha \beta = \frac{4}{2} = 2 \]

### Step 4: 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 found:
\[ \alpha^2 + \beta^2 = \left(\frac{3}{2}\right)^2 - 2 \cdot 2 \]

### Step 5: Calculate \( \left(\frac{3}{2}\right)^2 \)
Calculating \( \left(\frac{3}{2}\right)^2 \):
\[ \left(\frac{3}{2}\right)^2 = \frac{9}{4} \]

### Step 6: Calculate \( 2 \cdot 2 \)
Calculating \( 2 \cdot 2 \):
\[ 2 \cdot 2 = 4 \]

### Step 7: Substitute back into the identity
Now substituting back into the equation:
\[ \alpha^2 + \beta^2 = \frac{9}{4} - 4 \]

### Step 8: Convert 4 to a fraction with a common denominator
Convert \( 4 \) to a fraction:
\[ 4 = \frac{16}{4} \]

### Step 9: Perform the subtraction
Now perform the subtraction:
\[ \alpha^2 + \beta^2 = \frac{9}{4} - \frac{16}{4} = \frac{9 - 16}{4} = \frac{-7}{4} \]

### Final Answer
Thus, \( \alpha^2 + \beta^2 = -\frac{7}{4} \).

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