If `alpha` and `beta` are the roots of `x^(2)-3px+p^(2)=0` such that `alpha^(2)+beta^(2)=(7)/(4)` then values of p are

Options

A. 2, 1
B. `2, (1)/(2)`
C. `(1)/(2),1`
D. `(1)/(2), -(1)/(2)`

Answer

To solve the problem, we need to find the values of \( p \) given that \( \alpha \) and \( \beta \) are the roots of the quadratic equation \( x^2 - 3px + p^2 = 0 \) and that \( \alpha^2 + \beta^2 = \frac{7}{4} \).

### Step-by-step Solution:

1. **Identify the coefficients of the quadratic equation:**
The given quadratic equation is \( x^2 - 3px + p^2 = 0 \).
Here, \( a = 1 \), \( b = -3p \), and \( c = p^2 \).

2. **Use Vieta's formulas to find the sum and product of the roots:**
According to Vieta's formulas:
- The sum of the roots \( \alpha + \beta = -\frac{b}{a} = -\frac{-3p}{1} = 3p \).
- The product of the roots \( \alpha \beta = \frac{c}{a} = \frac{p^2}{1} = p^2 \).

3. **Express \( \alpha^2 + \beta^2 \) in terms of \( p \):**
We know that:
\[ \alpha^2 + \beta^2 = (\alpha + \beta)^2 - 2\alpha\beta \]
Substituting the values from Vieta's formulas:
\[ \alpha^2 + \beta^2 = (3p)^2 - 2(p^2) = 9p^2 - 2p^2 = 7p^2 \]

4. **Set up the equation based on the given condition:**
We are given that \( \alpha^2 + \beta^2 = \frac{7}{4} \). Therefore, we can set up the equation:
\[ 7p^2 = \frac{7}{4} \]

5. **Solve for \( p^2 \):**
Divide both sides by 7:
\[ p^2 = \frac{1}{4} \]

6. **Take the square root to find \( p \):**
Taking the square root of both sides gives:
\[ p = \pm \frac{1}{2} \]

### Final Answer:
The values of \( p \) are \( \frac{1}{2} \) and \( -\frac{1}{2} \).

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