. . Irrational Roots If we further restrict the coefficients of the quadratic equation ax bx = to be rational, we get some interesting results. Let us consider a quadratic equation ax bx with a , b , and c rational.
As usual let ∆= ac and let r and r be the roots. In this case, when ∆= , we have ; this root is not only real, it is in fact a rational number. When D is positive, then no doubt that D exists in and we get two distinct real roots. But D will be a rational number for certain values of a b , , and c , and it is an irrational number for other values of a b , , and c .
If D is rational, then both r and r are rational. If D is irrational, then both r and r are irrational. Immediately we have a question. If ∆> , when will D be rational and when will it be irrational?
To answer this question, first we observe that D is rational, as the coefficients are rational numbers. So ∆= m n for some positive integers m and n with m n ) = where m n ) denotes the - - Theory of Equations greatest common divisor of m and n . It is now easy to understand that D is rational if and only if both m and n are perfect squares. Also, D is irrational if and only if at least one of m and n is not a perfect square.
We are familiar with irrational numbers of the type p q where p and q are rational numbers q is irrational. Such numbers are called surds . As in the case of imaginary roots, we can prove that if p q is a root of a polynomial, then p q is also a root of the same polynomial equation, when all the coefficients are rational numbers. Though this is true for polynomial equation of any degree