if p and q are non zero constants, the equation `x^2+px+q=0` has roots ` alpha` and `beta` then the equation `qx^2+px+1=0` has roots

Options

A. `qx^(2) + px + 1 = 0 "has roots"(1)/(u)and(1)/(v)`
B. `(x-p)(x+q) = 0 "has roots" u + v and uv`
C. `x^(2) + p^(2) x + q^(2) = 0 "has roots" u^(2) and v^(2)`
D. `x^(2) + qx + p = 0"has roots" (u)/(v) and (v)/(u)`

Answer

To solve the problem step by step, we start with the given quadratic equation and its roots.

### Step 1: Understand the given quadratic equation
We have the quadratic equation:
\[ x^2 + px + q = 0 \]
with roots \( \alpha \) and \( \beta \).

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

### Step 3: Write the second quadratic equation
We need to analyze the second quadratic equation:
\[ qx^2 + px + 1 = 0 \]

### Step 4: Substitute the values of \( p \) and \( q \)
From the first equation, we have:
- \( p = -(\alpha + \beta) \)
- \( q = \alpha \beta \)

Substituting these into the second equation gives:
\[ (\alpha \beta)x^2 - (\alpha + \beta)x + 1 = 0 \]

### Step 5: Rewrite the equation in standard form
This can be rearranged as:
\[ \alpha \beta x^2 - (\alpha + \beta)x + 1 = 0 \]

### Step 6: Identify the new roots
Now we can express this quadratic equation in terms of its roots. The roots of the quadratic equation \( ax^2 + bx + c = 0 \) can be found using Vieta's formulas:
- The sum of the roots \( r_1 + r_2 = -\frac{b}{a} \)
- The product of the roots \( r_1 r_2 = \frac{c}{a} \)

For our equation:
- \( a = \alpha \beta \)
- \( b = -(\alpha + \beta) \)
- \( c = 1 \)

Thus, we have:
- The sum of the roots \( r_1 + r_2 = \frac{\alpha + \beta}{\alpha \beta} \)
- The product of the roots \( r_1 r_2 = \frac{1}{\alpha \beta} \)

### Step 7: Find the roots
The roots can be expressed as:
\[ r_1 = \frac{1}{\alpha} \quad \text{and} \quad r_2 = \frac{1}{\beta} \]

### Conclusion
Thus, the roots of the equation \( qx^2 + px + 1 = 0 \) are:
\[ \frac{1}{\alpha} \text{ and } \frac{1}{\beta} \]

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