the roots of 2x^2 - 3x = 4 are a and b. find the simplest quadratic equation which has roots 1/a and 1/b

Answer 1

the sum of the roots of the original is 3/2
the product of the roots is -2

so a+b = 3/2
ab = -2

sum of new roots = 1/a + 1/b = (a+b)/(ab)
= (3/2) / -2 = -3/4

product of new roots - (1/a)(1/b) = 1/(ab) = -1/2

so the new equation is
x^2 + (3/4)x - 1/2 = 0 or
4x^2 + 3x - 2 = 0

based on the theorem that for
ax^2 + bx + c = 0

the sum of the roots is -b/a and
the product of the roots is c/a