Answer to Question #171677 in Algebra for Mandy

Question

The roots of the equation 3x² + 4x – 5 = 0 are α, β. Find the values of the following:

1. 1/α + 1/β
2. α² + β²    

Answer 1

$$3x^2+4x-5=0$$

$a=3, b=4, c=-5$

By Vieta theorem

$$\alpha+\beta=-\dfrac{b}{a}$$

$$\alpha\beta=\dfrac{c}{a}$$

1)

$$\dfrac{1}{\alpha}+\dfrac{1}{\beta}=\dfrac{\alpha+\beta}{\alpha\beta}$$

$$=\dfrac{-\dfrac{b}{a}}{\dfrac{c}{a}}=-\dfrac{b}{c}=\dfrac{4}{5}$$

2)

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

$$=(-\dfrac{b}{a})^2-2(\dfrac{c}{a})$$

$$=(-\dfrac{4}{3})^2-2(\dfrac{-5}{3})=\dfrac{46}{9}$$