Discriminant - Nature of The Roots - Quadratic

Quadratic

Because the quadratic formula expressed the roots of a quadratic polynomial as a rational function in terms of the square root of the discriminant, the roots of a quadratic polynomial are in the same field as the coefficients if and only if the discriminant is a square in the field of coefficients: in other words, the polynomial factors over the field of coefficients if and only if the discriminant is a square.

Thus in particular for a quadratic polynomial with real coefficients, a real number has real square roots if and only if it is nonnegative, and these roots are distinct if and only if it is positive (not zero). Thus

• Δ > 0: 2 distinct real roots: factors over the reals;
• Δ < 0: 2 distinct complex roots (complex conjugate), does not factor over the reals;
• Δ = 0: 1 real root with multiplicity 2: factors over the reals as a square.

Further, for a quadratic polynomial with rational coefficients, it factors over the rationals if and only if the discriminant – which is necessarily a rational number, being a polynomial in the coefficients – is in fact a square.