Answer to Question #171677 in Algebra for Mandy

Question #171677

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    
1
Expert's answer
2021-03-16T08:40:17-0400
3x2+4x5=03x^2+4x-5=0

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

By Vieta theorem


α+β=ba\alpha+\beta=-\dfrac{b}{a}

αβ=ca\alpha\beta=\dfrac{c}{a}

1)


1α+1β=α+βαβ\dfrac{1}{\alpha}+\dfrac{1}{\beta}=\dfrac{\alpha+\beta}{\alpha\beta}

=baca=bc=45=\dfrac{-\dfrac{b}{a}}{\dfrac{c}{a}}=-\dfrac{b}{c}=\dfrac{4}{5}

2)


α2+β2=(α+β)22αβ\alpha^2+\beta^2=(\alpha+\beta)^2-2\alpha\beta

=(ba)22(ca)=(-\dfrac{b}{a})^2-2(\dfrac{c}{a})

=(43)22(53)=469=(-\dfrac{4}{3})^2-2(\dfrac{-5}{3})=\dfrac{46}{9}



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