If the sum of the roots of the equation `x^(2)+px+q=0` is 3 times their difference, then

Options

A. a) `p^(2)= 6q`
B. b) `2p^(2)= 9q`
C. c) `4p^(2)= 14q`
D. d) `p^(2)+q^(2)=4 `

Answer

To solve the problem, we need to analyze the given quadratic equation and the condition provided about the roots. Let's break it down step by step.

### Step 1: Identify the roots and their properties
Let the roots of the quadratic equation \( x^2 + px + q = 0 \) be \( \alpha \) and \( \beta \).

### Step 2: Use the sum and product of roots
From Vieta's formulas, we know:
- The sum of the roots \( \alpha + \beta = -\frac{p}{1} = -p \) (Equation 1)
- The product of the roots \( \alpha \beta = \frac{q}{1} = q \) (Equation 2)

### Step 3: Set up the condition given in the problem
The problem states that the sum of the roots is 3 times their difference:
\[ \alpha + \beta = 3(\alpha - \beta) \]

### Step 4: Substitute the sum of roots into the condition
Substituting Equation 1 into the condition gives:
\[ -p = 3(\alpha - \beta) \]

### Step 5: Express the difference of the roots
We can express \( \alpha - \beta \) in terms of \( \alpha + \beta \) and \( \alpha \beta \):
\[ \alpha - \beta = \sqrt{(\alpha + \beta)^2 - 4\alpha\beta} \]
Substituting from Equations 1 and 2:
\[ \alpha - \beta = \sqrt{(-p)^2 - 4q} = \sqrt{p^2 - 4q} \]

### Step 6: Substitute the difference back into the condition
Now substituting this into the condition:
\[ -p = 3\sqrt{p^2 - 4q} \]

### Step 7: Square both sides to eliminate the square root
Squaring both sides gives:
\[ p^2 = 9(p^2 - 4q) \]

### Step 8: Simplify the equation
Expanding the right side:
\[ p^2 = 9p^2 - 36q \]
Rearranging gives:
\[ p^2 - 9p^2 + 36q = 0 \]
\[ -8p^2 + 36q = 0 \]
\[ 8p^2 = 36q \]
Dividing both sides by 4:
\[ 2p^2 = 9q \]

### Final Result
Thus, we have derived the relationship:
\[ 2p^2 = 9q \]

### Conclusion
The final answer is \( 2p^2 = 9q \).

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