If the sum of the roots of the quadratic equations `ax^(2) + bx+c=0` is equal to the sum of the squares of their reciprocals, then be `(b^2)/(ac) + (bc)/(a^2) =`

Options

A. 0
B. `-1`
C. 1
D. 2

Answer

To solve the problem, we need to establish the relationship between the sum of the roots of the quadratic equation and the sum of the squares of their reciprocals.

Given the quadratic equation:

\[ ax^2 + bx + c = 0 \]

Let the roots be \( \alpha \) and \( \beta \).

### Step 1: Find the sum of the roots
The sum of the roots \( \alpha + \beta \) can be expressed using Vieta's formulas:

\[ \alpha + \beta = -\frac{b}{a} \]

### Step 2: Find the product of the roots
The product of the roots \( \alpha \beta \) is given by:

\[ \alpha \beta = \frac{c}{a} \]

### Step 3: Find the sum of the squares of the reciprocals
The sum of the squares of the reciprocals of the roots is given by:

\[ \frac{1}{\alpha^2} + \frac{1}{\beta^2} \]

This can be rewritten using the identity:

\[ \frac{1}{\alpha^2} + \frac{1}{\beta^2} = \frac{\beta^2 + \alpha^2}{\alpha^2 \beta^2} \]

Using the identity \( \alpha^2 + \beta^2 = (\alpha + \beta)^2 - 2\alpha\beta \):

\[ \alpha^2 + \beta^2 = \left(-\frac{b}{a}\right)^2 - 2\left(\frac{c}{a}\right) = \frac{b^2}{a^2} - \frac{2c}{a} \]

Thus,

\[ \frac{1}{\alpha^2} + \frac{1}{\beta^2} = \frac{\frac{b^2}{a^2} - \frac{2c}{a}}{\left(\frac{c}{a}\right)^2} = \frac{b^2 - 2ac}{c^2} \]

### Step 4: Set the two expressions equal
According to the problem, we have:

\[ \alpha + \beta = \frac{1}{\alpha^2} + \frac{1}{\beta^2} \]

Substituting the values we found:

\[ -\frac{b}{a} = \frac{b^2 - 2ac}{c^2} \]

### Step 5: Cross-multiply to eliminate the fractions
Cross-multiplying gives:

\[ -bc^2 = a(b^2 - 2ac) \]

### Step 6: Rearranging the equation
Expanding the right side:

\[ -bc^2 = ab^2 - 2a^2c \]

Rearranging gives:

\[ ab^2 + bc^2 - 2a^2c = 0 \]

### Step 7: Divide through by \( ac \)
Dividing the entire equation by \( ac \):

\[ \frac{b^2}{ac} + \frac{bc}{a^2} - 2 = 0 \]

### Step 8: Solve for the desired expression
This simplifies to:

\[ \frac{b^2}{ac} + \frac{bc}{a^2} = 2 \]

Thus, the final answer is:

\[ \frac{b^2}{ac} + \frac{bc}{a^2} = 2 \]