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!

 

[rwisz]

The condition for the equation px^2 + qx + r = 0 to have two different real roots is that the discriminant, b^2 - 4ac, must be greater than 0. This means that q^2 - 4pr > 0. This condition ensures that the equation has two distinct solutions when solved for x.

The condition for the equation to have two equal real roots is that the discriminant, b^2 - 4ac, must be equal to 0. This means that q^2 - 4pr = 0. This condition ensures that the equation has only one solution, which is repeated, when solved for x.

The condition for the equation to have no real roots is that the discriminant, b^2 - 4ac, must be less than 0. This means that q^2 - 4pr < 0. This condition ensures that the equation has no real solutions when solved for x. In this case, the solutions would be complex numbers.

I hope this helps clarify the conditions for the equation to have different types of roots. If you are still unsure, I would recommend consulting with your teacher or a classmate for further explanation and examples.