If the equation ` 2 x^(2) - 7 x + 12 = 0` has two roots ` alpha and beta ` then the value of ` (alpha)/( beta) + (beta)/( alpha)` is

Options

A. `(7)/(2)`
B. `(1)/(24)`
C. `(7)/(24)`
D. `(97)/(24)`

Answer

To solve the equation \( 2x^2 - 7x + 12 = 0 \) and find the value of \( \frac{\alpha}{\beta} + \frac{\beta}{\alpha} \), where \( \alpha \) and \( \beta \) are the roots of the equation, we can follow these steps:

### Step 1: Identify the coefficients
The given quadratic equation is in the standard form \( ax^2 + bx + c = 0 \), where:
- \( a = 2 \)
- \( b = -7 \)
- \( c = 12 \)

### Step 2: Use Vieta's formulas
According to Vieta's formulas:
- The sum of the roots \( \alpha + \beta = -\frac{b}{a} \)
- The product of the roots \( \alpha \beta = \frac{c}{a} \)

Calculating these values:
- \( \alpha + \beta = -\frac{-7}{2} = \frac{7}{2} \)
- \( \alpha \beta = \frac{12}{2} = 6 \)

### Step 3: Find \( \frac{\alpha}{\beta} + \frac{\beta}{\alpha} \)
We can express \( \frac{\alpha}{\beta} + \frac{\beta}{\alpha} \) in terms of \( \alpha + \beta \) and \( \alpha \beta \):
\[ \frac{\alpha}{\beta} + \frac{\beta}{\alpha} = \frac{\alpha^2 + \beta^2}{\alpha \beta} \]

### Step 4: Calculate \( \alpha^2 + \beta^2 \)
Using the identity \( \alpha^2 + \beta^2 = (\alpha + \beta)^2 - 2\alpha\beta \):
\[ \alpha^2 + \beta^2 = \left(\frac{7}{2}\right)^2 - 2 \times 6 \]
Calculating \( \left(\frac{7}{2}\right)^2 \):
\[ \left(\frac{7}{2}\right)^2 = \frac{49}{4} \]
Calculating \( 2 \times 6 = 12 \) (which can be expressed with a common denominator):
\[ 12 = \frac{48}{4} \]
Now substituting back:
\[ \alpha^2 + \beta^2 = \frac{49}{4} - \frac{48}{4} = \frac{1}{4} \]

### Step 5: Substitute into the expression
Now substituting \( \alpha^2 + \beta^2 \) and \( \alpha \beta \) into the expression:
\[ \frac{\alpha^2 + \beta^2}{\alpha \beta} = \frac{\frac{1}{4}}{6} = \frac{1}{4} \times \frac{1}{6} = \frac{1}{24} \]

### Final Answer
Thus, the value of \( \frac{\alpha}{\beta} + \frac{\beta}{\alpha} \) is \( \frac{1}{24} \).

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