Question

If a, b ,c are rational and a +b+ c=0, prove that the roots of the
Quadratic equation ax^2+bx+ c =0 are rational.

Answer 1

a + b + c = 0
b = - a - c

b = -(a + c)

Discriminant of ax² + bx + c is given by

D = b² - 4ac = { - (a + c)²} - 4ac (Putting value of b)

= (a + c)² - 4ac = a² + c² + 2ac - 4ac

= a² + c² - 2ac = (a - c)² ( A perfect square)

Thus roots = (- b ± √D)/2a = {-b ± √(a - c)²}/2a = (-b + a - c/2a) ( A rational number as a, b, c are rational numbers)

Thus roots = rational.

Thus we can conclude that if discriminant is a perfect square then roots are rational.