Show that the roots of x^2 + x =8 are irrational

Question

Prove the equation is irrational

Answer 1

x^2 + x = 8

or

x^2 + x -8 = 0

There are 2 cases where the roots can be irrational for any real valued quadratic equation of the form

ax^2 + bx +c = 0

1) If D = b^2 - 4ac > 0 is a perfect square and either a or b is irrational. But in our case a = 1 and b = 1, so this case do not apply.

2) If D = b^2 - 4ac>0 is not a perfect square. We need to investigate this,

given a =1, b = 1 and c= -8

so, D = 1 - 4*(-8) = 1+32 = 33

and sqrt(33) is not a perfect square. Therefore the roots of the given quadratic equation is irrational.