Math Is Fun Forum

Identity

Member

Registered: 2007-04-18

Posts: 934

Please check my reasoning

Could someone please check my reasoning with this question? I'm not sure about it

Question:
Find the values of

for

for which no real solutions exist.

Let  then

.
If you look at the graph of

, it is a parabola with a minimum. You must make it so that either:
1. The graph does not touch the a-axis or
2. The graph only has non-positive roots for a.
Note that the term  translates the graph left or right.
For 1.

, so solving,

.
For 2.

Assuming

is positive (which must be true for a negative translation and hence negative roots) and considering only the right root,

Since all these steps can be done in reverse (i.e. go from

to

, then the roots will be negative if and only if

and b is positive, which is true.
Hence, the total set of values for which no solutions exist is

.

Last edited by Identity (2008-10-10 19:56:20)

mathsyperson

Moderator

Registered: 2005-06-22

Posts: 4,900

Re: Please check my reasoning

That all looks good, except you forgot to include case 1 in your final result.

So actually

---

Why did the vector cross the road?
It wanted to be normal.

Identity

Member

Registered: 2007-04-18

Posts: 934

Re: Please check my reasoning

ok thanks mathsy

I was mostly worried about my 'reverse working' of the inequality being some kind of fallacy