# 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

### Dan

#### Latest posts by Dan (see all)

- 2016 MMC Schedule - November 4, 2015
- 2014 MTAP reviewer for Grade 3 - September 30, 2015
- 2015 MTAP reviewer for 4th year solution part 1 - August 22, 2015

Ive by no means read something like this before. So good to find somebody with some original thoughts on this topic, really thank you for beginning this up. this web site is one thing that is necessary on the net, a person having a little originality. beneficial job for bringing some thing new to the web!

Great blog! Do you have any recommendations for aspiring writers? I’m hoping to start my own blog soon but I’m a little lost on everything. Would you suggest starting with a free platform like WordPress or go for a paid option? There are so many options out there that I’m completely confused .. Any ideas? Bless you!

C31EYg Just Browsing While I was surfing today I saw a excellent article concerning