​If α and β where α > β. are the roots of the equation 5x² – 3x – 2 = 0, then what is the value of (α – 2β) ?

Correct option is A

​Given:
Equation: $$5x^2 - 3x - 2 = 0$$​​

Formula Used: 

For a quadratic equation $$ax^2 + bx + c = 0,$$​​
Sum of roots (α+β) = $$-\frac {b}{a}$$​, Product of roots (αβ) = $$\frac {c}{a}$$ 

​$$(\alpha - \beta)^2 = (\alpha + \beta)^2 - 4\alpha \beta$$ 

$$\alpha - 2\beta = (\alpha - \beta) - \beta$$​​

Solution: ​

​$$5x^2 - 3x - 2 = 0$$​​
here, a = 5, b = -3, c = -2: 

So, 

​$$\alpha + \beta = \frac{-(-3)}{5} = \frac{3}{5}, \quad \alpha \beta = \frac{-2}{5}$$​​

​$$(\alpha - \beta)^2 = \left( \frac{3}{5} \right)^2 - 4\left( -\frac{2}{5} \right)$$​​

​$$= \frac{9}{25} + \frac{8}{5}$$​​

​$$= \frac{49}{25}$$​​

​$$\implies \alpha - \beta = \frac{7}{5} \quad (\text{since } \alpha > \beta)$$​​

Using identity $$\alpha - 2\beta = (\alpha - \beta) - \beta$$​:

From $$\alpha + \beta = \frac{3}{5}$$​ and $$\alpha - \beta = \frac{7}{5}$$​,

​$$2\beta = (\alpha + \beta) - (\alpha - \beta)$$​​

​$$= \frac{3}{5} - \frac{7}{5}$$​​

​$$= -\frac{4}{5}\Rightarrow \beta = -\frac{2}{5}$$​​

Now,

​$$\alpha - 2\beta = \frac{7}{5} - \left( -\frac{2}{5} \right) = \frac{9}{5}$$​​