If a and ß are the roots of the $2x^{2}-3x-6=0$ then $\frac{1}{\alpha}+\frac{1}{\beta}=?$

Question

Accepted Answer

The correct option is 1:
To solve this problem, we need to recall the relationship between the roots and coefficients of a quadratic equation.

Key Concepts:
For a quadratic equation in the standard form $ax^2 + bx + c = 0$, if $\alpha$ and $\beta$ are its roots, then:1.  Sum of the roots: $\alpha + \beta = -\frac{b}{a}$
2.  Product of the roots: $\alpha \beta = \frac{c}{a}$Given the quadratic equation: $2x^2 - 3x - 6 = 0$
Comparing this with the standard form $ax^2 + bx + c = 0$, we have:
$a = 2$
$b = -3$
$c = -6$

Step-by-step explanation:
First, let's find the sum and product of the roots ($\alpha$ and $\beta$) using the formulas.
Sum of the roots:
$\alpha + \beta = -\frac{b}{a} = -\frac{(-3)}{2} = \frac{3}{2}$
Product of the roots:
$\alpha \beta = \frac{c}{a} = \frac{-6}{2} = -3$

Now, we need to find the value of $\frac{1}{\alpha} + \frac{1}{\beta}$.
To add these fractions, we find a common denominator, which is $\alpha \beta$.
$\frac{1}{\alpha} + \frac{1}{\beta} = \frac{\beta}{\alpha \beta} + \frac{\alpha}{\alpha \beta} = \frac{\alpha + \beta}{\alpha \beta}$
Now, substitute the values of $(\alpha + \beta)$ and $(\alpha \beta)$ that we calculated:
$\frac{\alpha + \beta}{\alpha \beta} = \frac{\frac{3}{2}}{-3}$
To simplify this expression, we can multiply the numerator by the reciprocal of the denominator:
$\frac{3}{2} 	imes \frac{1}{-3} = \frac{3}{-6} = -\frac{1}{2}$

Thus, the value of $\frac{1}{\alpha} + \frac{1}{\beta}$ is $-\frac{1}{2}$.

The correct option is 1, which states $\frac{-1}{2}$. This matches our calculated result, confirming that option 1 is correct.

The incorrect option is 2: 0
This option would be correct if $\alpha + \beta = 0$, which means the sum of the roots is zero. In this equation, the sum of the roots is $\frac{3}{2}$, not 0.

The incorrect option is 3: 1
This option is incorrect. A common mistake might be to assume a simpler relationship or make an arithmetic error during substitution or simplification.

The incorrect option is 4: $\frac{3}{2}$
This option is incorrect. $\frac{3}{2}$ is the sum of the roots $(\alpha + \beta)$, not the value of $\frac{1}{\alpha} + \frac{1}{\beta}$.