​If $$α,β$$​ are the roots of $$px² + qx + r = 0$$ then $$α^2+β^2=$$.....?

Correct option is B

Given:

​$\alpha$ and $\beta$ are the roots of the quadratic equation: $$px^2 + qx + r = 0$$ 

$$\alpha^2 + \beta^2 = ?$$ 

Formula Used: ​

​$$(a +b)^2 = a^2+b^2+2ab$$​​

Solution:

For the quadratic equation $$px^2 + qx + r = 0$$ the sum and product of the roots are: 

Sum of roots = $$\alpha + \beta = -\frac{q}{p}$$ 

Product of roots = $$\alpha \beta = \frac{r}{p}$$ 

​We use the identity:

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

Substitute the expressions for α + β and αβ:

​$$\alpha^2 + \beta^2 = \left(-\frac{q}{p}\right)^2 - 2\left(\frac{r}{p}\right)$$ 

$$\alpha^2 + \beta^2 = \frac{q^2}{p^2} - \frac{2r}{p}$$ 

$$\alpha^2 + \beta^2 = \frac{q^2-2pr}{p^2}$$ 

Thus, the expression for $$\alpha^2 + \beta^2$$ is :  $$\frac{(q^2-2pr)}{p^2}$$ .​