Finding a minimum / polynomials / complex numbers

[res publica]

Can anybody please tell me how to find correct values for c in the following equation? The equation has complex roots.

1) Find the values of x for cx² + 2x - (c+1) = 0

2) e^(2x) + 2e^x is bigger or equal to 3e^(x) + 6

3) f(x) = |x+5| + |x+2| + |x| + |x-3|

The equation gives us the total length of a cable between the points A and B

P=-5
Q= -2
O=0
R= 3

x is the centre of the cable.

How can I graph this function that has absolute values in its equation?

The question is, "Where should the centre of the cable be to minimize toe total length of cable to all four points? What is the minimum (of the length of the cable)?"

Last part of this problem: A point S is added 7km away from O (so it adds to the equation + |x-7|. Where should it be for a minimal cable length?

4) I need to find a fourth degree polynomial with the rational coefficients 2- i*squareroot 3 and squareroot2 +1 as two of the four zeros. How do I do this and what is the solution? Please help me. Thanks a lot.

 

[o_O]

1. Think about the properties of the discriminant:
If \(\displaystyle b^{2} - 4ac > 0\), the parabola has 2 real roots.
If \(\displaystyle b^{2} - 4ac = 0\), the parabola has 1 real root.
If \(\displaystyle b^{2} - 4ac < 0\), the parabola has no real roots.

2. You mean solve for x? Move everything to the left side and factor:
\(\displaystyle e^{2x} +2e^{x} - 3e^{x} - 6 \geq 0\)
\(\displaystyle e^{2x} - e^{x} - 6 \geq 0\)

Got to jet. Maybe someone else will help you out with the rest!

 

[stapel]

3) I have no idea what's going on with this one. How do P, Q, etc, relate to the rest of it?

4) Use what you know, from the Quadratic Formula, about roots with radicals. This will allow you to find all four roots. Then use the fact that "x = a is a root" means "x - a is a factor". Multiply the four factors to find the polynomial.

:wink:

Eliz.

 

[res publica]

Well thanks so far. But I don't know how to solve these inequalities. Please someone tell me and explain why it is the way it is! For the first one I got k2+k<1. But what then? How to continue? The same for the second one, what to d? Please help me.

 

[pka]

But I don't know how to solve these inequalities.

Please tell us why you were given this problem to do if you know nothing about solving inequalities. That seems odd.

 

[res publica]

But I don't know how to solve these inequalities.

Please tell us why you were given this problem to do if you know nothing about solving inequalities. That seems odd.

Well, I don't know. We're always working on a number of totally different topics, maybe it's assumed that I know how to solve them, but unfortunately I don't. Can you please help me?

 

[o_O]

Here's a link that goes over inequalities involving quadratics:
http://www.purplemath.com/modules/ineqquad.htm

Probably help you answer the first two questions.