If α and β are the roots of the equation $$6x^2+x-15=0$$​, where α>β, then the value of (α-β) is:

Correct option is B

Given:  

$$\alpha$$ and $$\beta$$ are the roots of the equation $$6x^2 + x - 15 = 0$$ . Where $$\alpha > \beta$$​​

Concept Used:

For a quadratic equation of the form $$ax^2 + bx + c = 0$$ the relationships between the sum and product of the roots are given by:

Sum of the roots $$\alpha + \beta = -\frac{b}{a}$$ 

Product of the roots $$\alpha \beta = \frac{c}{a}$$ 

Solution:

Let α and β be the roots of this quadratic equation.

For the equation $$6x^2 + x - 15 = 0$$​  the coefficients are: ​

a = 6 , b = 1 , c = −15 

Thus, we calculate:

​$$\alpha + \beta = -\frac{1}{6}$$ 

$$\alpha \beta = \frac{-15}{6} = -\frac{5}{2}$$ 

To find α − β, we use the following formula:

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

$$(\alpha - \beta)^2 = \left(-\frac{1}{6}\right)^2 - 4\left(-\frac{5}{2}\right)$$ 

$$(\alpha - \beta)^2 = \frac{1}{36} + 10 = \frac{1+360}{36} = \frac{361}{36}$$ 

$$\alpha - \beta = \sqrt{\frac{361}{36}} = \frac{19}{6}$$ 

Thus the value of $$\alpha-\beta$$ is $$\frac{19}{6}$$.​

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