What Determines the Number of Real Roots in a Quadratic Equation?

[Hyzon]

Homework Statement

State the condition for the equation px^2 + qx + r = 0 to have:
a)two different real roots
b)two equal real roots
c)no real roots

Homework Equations

b^2-4ac maybe?

The Attempt at a Solution

I missed a class due to ailments and now have to catch up on missed work. I can't find what the question means by state the condition. All I need to know is what that even means, and maybe where to start working.

I've tried using the discriminant of the equation but I don't know what to do because the coefficients are variables, even the constant is a variable. Any help would be great, am I even on the right track with using discriminants?

I just tried using the discriminant of the equation with variables and for each a,b,c I came out with q^2-4pr > 0, q^2-4pr=0, q^2-4pr<0 respectively. I'm not sure if that is correct though.

 

[1994Bhaskar]

NO PROBLEM hyzon!
BY THE WAY WHAT YOU ARE THINKING IS CORRECT!YES THAT DISCRIMINANT WILL CERTAINLY HELP YOU & YOU ARE IN THE RIGHT WAY!
FOR ANY QUADRATIC EQUATION px²+qx+r=0
we have solution of x=(-q(+or-)squareroot(q²-4pr))/(2*p)
WELL IT IS MORE POPULAR AS b^2-4a*c.
a)for roots to be real we just want the term inside the root to be greater than or equal to zero.
=>b^2-4a*c>=0
OR HERE IN THIS CASE:
q^2-4pr>=0
=>q²>=4pr is the required condition for real roots.
IF THIS IS NOT SO THEN THE ROOT'S ARE NOT REAL AND ARE THUS IMAGINARY.
THUS a) AND c) ARE DONE.
b)HERE b^2-4ac=0 is condition for real roots.
=>q^2-4pr=0
=>q²=4pr is the required condition for equal roots.

 

[Hyzon]

Thanks, yeah so I get that now. But I'm stumped again on the same topic.

For what values of k does each equation have two different real roots?
a) x2+kx+1=0

I used the discriminant

k^2-4(1)(1)
k^2-4
k-2
k>2, k<(-2) or |k|>2

I understand that one, but question b) has k in the a position not b.

b)kx^2+4x-3=0

I put it in discriminant form

4^2-4(k)(-3)

Now what? How do I find the values of k to have two different real roots?

 

[epenguin]

You use the condition discriminant > 0 like before, only it's a bit simpler to get k!