Couple of algebra questions I'm struggling with

[MadmanMurray]

Homework Statement

1.)
a.) If the roots of equation px² - 4x + 1 are equal then find value of p
b.) Prove that the equation (k - 2)x² + 2x - k = 0 has real roots (upside down A symbol) k E R

Homework Equations

The Attempt at a Solution

I just don't know how to approach these questions. For a.) it says roots are equal meaning there's only 1 root but how do I find out what that root is if I have a constant in there?

For b.) I have the same problem. I would usually plug an equation into the quadratic formula to find out if it has real roots but in this case there's a constant so I'm lost. Do I have to find this constant first and if so how?

 

[snipez90]

I'm pretty sure you just need to consider the discriminant.

For a) What does the discriminant have to be for the two roots to be equal (hint: look at the quadratic equation)

For b) Compute the discriminant

 

[MadmanMurray]

Ah yea I didn't think about the fact it has to come to 0 thanks a lot. Can you verify my answer real quick I got P = 4 for a.)

For b.) do I need to find out what k is first?

 

[Dick]

Yes, p=4. For the second one look at the discriminant again. You can't find k. You just need to show that the discriminant is nonnegative for all k.

 

[snipez90]

Yes, p = 4 is correct. You should verify it by factoring the the expression and checking to see that there is only one root.

For b), no, the question suggests that k is any real number. The idea is to show that no matter what k is, the roots of the resulting quadratic are real.

Remember, the discriminant is the quantity under the square root. What should the sign of the discriminant be?

 

[MadmanMurray]

Ah right that clears it up thanks a lot for the help.