Given the equation 2x2 5x6 0 Is it possible without solving

Given the equation 2x^2 - 5x-6 0 Is it possible without solving the equation to find the sum of the reciprocals of its solutions? Is it possible without solving the equation to build a new quadratic equation whose solutions are greater than the solutions of this equation by 1.

Solution

2x^2 - 5x -6 =0

i) for ax^2 +bx +c ; sum of roots: x1 +x2 = -b

x1 +x2 = 5

Reciprocal of sum of roots = 1/(x1 +x2) = 1/5

ii) Equation with roots 1 greater than x which is root of the given equation

Plug x = x +1 in equation.

2(x+1)^2 - 5(x +1) -6 = 2(x^2 +1 +2x) - 5x -5 -6

= 2x^2 -x - 9