# Natures of the Roots of Quadratic Equations

Discriminant is the most important part of quadratic functions when we talk about the nature of its roots. Given a quadratic equation , discriminant (D) can be defined mathematically as .

Discriminant Nature of Roots

Equal

Real and distinct

Imaginary Roots

Let me also show you the graph.

If D=0, it belongs to the family of perfect square trinomials in the form of and has two the same roots.

If D>0, it belongs to the parabola that intersect the x-axis at two points

If D<0, it belongs to the parabola that the does not intersect the x-axis. That is why we call it imaginary because roots are the values of x that the given equation will intersect the x-axis. Does that make sense?

Worked Problem 1:

For what values of k does the equation to have equal roots?

Solution:

For the equation to have real roots,

By substitution:

a=3, b=2k, c=3

Worked Problem 2: 2007 MMC Division finals

For what values of k does the equation has real and distinct roots?

Solution:

By substitution:

a=3, b=k, c=3

If you don’t know how to solve this, better read first solving quadratic inequality.

Solving for k we have

Worked Problem 3:

For what values of k does the equation will have an imaginary roots.

Solution:

By substitution:

a=2, b=k, c=k-2

with multiplicity of 2

If we follow the rules in quadratic inequality the solution must be 2<x<2 but is there an interval that is greater than 2 and less than 2? The answer is all real numbers except 2. Or we can simply say