What is the Discriminant?
The discriminant indicated normally by
Given a second degree equation in the general form:
the discriminant is:
The discriminant can be used to characterize the solutions of the equation as:
The discriminant can also come in handy when attempting to factorize quadratics. If
I hope that helps!
The discriminant of a polynomial equation is a value computed from the coefficients which helps us determine the type of roots it has - specifically whether they are real or non-real and distinct or repeated.
#ax^2+bx+c = 0#
is given by the formula:
#Delta = b^2-4ac#
From the discriminant we can discriminate whether the equation has two real roots, one repeated real root or two non-real roots.
#Delta > 0#then the quadratic equation has two distinct real roots.
#Delta = 0#then the quadratic equation has one repeated real root.
#Delta < 0#then the quadratic equation has no real roots. It has a complex conjugate pair of non-real roots.
For a cubic equation with real coefficients in standard form:
#ax^3+bx^2+cx+d = 0#
#Delta = b^2c^2-4ac^3-4b^3d-27a^2d^2+18abcd#
#Delta > 0#then the cubic equation has three real roots.
#Delta = 0#then the cubic has a repeated root. It may have one real root of multiplicity #3#. Otherwise it may have two distinct real roots, one of which is of multiplicity #2#.
#Delta < 0#then the cubic equation has one real root and a complex conjugate pair of complex roots.
Polynomial equations of higher degree also have discriminants, which help determine the nature of the roots, but they are less useful for quartics and above.
See https://socratic.org/s/aLqgSvFm for more details.