Question

If α and β are zeroes of 3x2 – 2x – 6, then find α

2 + β

2

.​

Answer 1

Explanation:

$\textsf{We know that in a quadratic polynomial of form ax^2 + bx + c}$$\textsf{We know that in a quadratic polynomial of form}$  $\mathsf{ax^2 + bx + c}$ $\textsf{with roots} \: \alpha \textsf{and} \: \beta,$

$\mathsf{\alpha +\beta=\frac{-b}{a}}\: \textsf {and} \: \mathsf{\alpha\beta=\frac{c}{a}}$

$\textsf{In the given polynomial, a = 3, b = -2 and c = -6, and we wish to find }\alpha^2 + \beta^2.$

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

$\mathsf{=(\frac{2}{3})^2- (2\times\frac{-6}{3}})$

$\mathsf{=\frac{4}{9}-(2\times-2)}$

$\mathsf{=\frac{4}{9}+4 = \frac{4+36}{9}}$

$\mathsf={\frac{40}{9}}$

Answer: 40/9