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

If a<0, Equal Roots

 

a>0

a>0. Equal Roots

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

Real and Distinct root

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

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

 

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