Question 939564
Find the exact value of the sum of the reciprocal of the roots of the equation 3x^2-4x-5=0
<pre>
{{{x = (- b +- sqrt(b^2 - 4ac))/(2a)}}}
With a being 3; b being - 4; and c being - 5, we get:
{{{x = (- - 4 +- sqrt((- 4)^2 - 4(3)(- 5)))/(2(3))}}}______{{{x = (4 +- sqrt(16 + 60))/6}}}______{{{x = (4 +- sqrt(76))/6}}}
{{{x = (4 +- sqrt(4 * 19))/6}}}______{{{x = (4 +- 2sqrt(19))/6}}}______{{{x = 2(2 +- sqrt(19))/6}}}______{{{x = cross(2)(2 +- sqrt(19))/3cross(6))}}}

Exact root 1: {{{(2 + sqrt(19))/3}}}, and exact root 1's reciprocal: {{{3/(2 + sqrt(19))}}}

Exact root 2: {{{(2 - sqrt(19))/3}}}, and exact root 2's reciprocal: {{{3/(2 - sqrt(19))}}}
Sum of exact roots' reciprocals: {{{(3/(2 + sqrt(19)) + 3/(2 - sqrt(19)))}}}
{{{(3(2 - sqrt(19)) + 3(2 + sqrt(19)))/(2 + sqrt(19))(2 - sqrt(19))}}} -------- Multiplying by LCD, {{{(2 + sqrt(19))(2 - sqrt(19))}}} 
{{{(6 - 3sqrt(19) + 6 + 3sqrt(19))/(2 + sqrt(19))(2 - sqrt(19))}}}_____{{{12/(2 + sqrt(19))(2 - sqrt(19))}}}______{{{12/(4 - 19)}}}______{{{12/(- 15)}}}______{{{4cross(12)/(- 5cross(15))}}}
Exact value of sum of exact roots' reciprocals: {{{highlight_green(- 4/5)}}}