# 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 $y=ax^2+bx+c$, discriminant (D) can be defined mathematically as $D=b^2-4ac$.

Discriminant                                                   Nature of Roots

$D=0$                                                               Equal

$D>0$                                                               Real and distinct

$D<0$                                                               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 $y=a(x-h)^2$ and has two the same roots.

If a<0, Equal Roots

a>0. Equal Roots

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

Real and Distinct root

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?

Imaginary roots

Worked Problem 1:

For what values of k does the equation $3x^2+2kx+3$ to have equal roots?

Solution:

For the equation to have real roots,

$b^2-4ac=0$

By substitution:

a=3,   b=2k,   c=3

$(2k)^2-4(3)(3)=0$ $4k^2=36$ $k=\pm 3$

Worked Problem 2: 2007 MMC Division finals

For what values of k does the equation $y=3x^2+kx+3$ has real and distinct roots?

Solution:

$b^2-4ac>0$

By substitution:

a=3,   b=k,   c=3

$k^2-4(3)(3)>0$

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

Solving for k we have

$k>6 or k<-6$

Worked Problem 3:

For what values of k does the equation $x^2+kx-1+k=0$ will have an imaginary roots.

Solution:

$b^2-4ac<0$

By substitution:

a=2,   b=k,   c=k-2

$k^2-4(1)(k-1)<0$ $k^2-4k+4<0$

$k=2$ 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 $x\ne 2$