quadratic equation

**alexandros** · June 5th 2010, 09:15 AM

Challenge problem:

Given that

with the equation having two unequal real roots,

Prove:

moderator approved CB

**Deadstar** · June 5th 2010, 11:09 AM

I've probably done something wrong but here goes...

Using standard formula we get roots to be

$\frac{2a \pm \sqrt{4a^2 - 4b}}{2} = a \pm \sqrt{a^2 - b}$ .

Since roots are real and unequal, $a^2 > b$

Hence...

$16a^4 + a^2 b^2 - 16a^2b + 2ab + 4b^2 + 3 > 16b^2 + a^2 b^2 - 16b^2 + 2ab + 4b^2 + 3$ .

Canceling we get

$= a^2 b^2 + 2ab + 4b^2 + 3 = (ab + 1)^2 - 1 + 3 + 4b^2 = (ab+1)^2 + 4b^2 + 2 \geq 2$ ...

Hence result.

**mr fantastic** · June 5th 2010, 03:13 PM

There are rules for posting in this particular subforum. Read them.

**CaptainBlack** · June 5th 2010, 03:47 PM

Originally Posted by mr fantastic

There are rules for posting in this particular subforum. Read them.

I have received the solution and said to post. But when I checked it had not been posted so I did not add the imprimatur.

Just shows the point of sending the solution to all the moderators.

CB

**mr fantastic** · June 5th 2010, 04:05 PM

Originally Posted by CaptainBlack

I have received the solution and said to post. But when I checked it had not been posted so I did not add the imprimatur.

Just shows the point of sending the solution to all the moderators. Mr F says: That is, to do what it says to do in the rules (clearly stated in the sticky) for posting in this subforum.

CB

..

**Deadstar** · June 5th 2010, 06:16 PM

Originally Posted by mr fantastic

Wow Deadstar what a Fantastic solution!

Originally Posted by CaptainBlack

Yes Mr F I agree! I believe Deadstar should be promoted to a mod!

CB

Originally Posted by mr fantastic

I don't think a mod is good enough... Administrator would be better.

Cheers guys!

**alexandros** · June 6th 2010, 04:50 PM

Originally Posted by Deadstar

I've probably done something wrong but here goes...

Using standard formula we get roots to be

$\frac{2a \pm \sqrt{4a^2 - 4b}}{2} = a \pm \sqrt{a^2 - b}$ .

Since roots are real and unequal, $a^2 > b$

Hence...

$16a^4 + a^2 b^2 - 16a^2b + 2ab + 4b^2 + 3 > 16b^2 + a^2 b^2 - 16b^2 + 2ab + 4b^2 + 3$ .

Canceling we get

$= a^2 b^2 + 2ab + 4b^2 + 3 = (ab + 1)^2 - 1 + 3 + 4b^2 = (ab+1)^2 + 4b^2 + 2 \geq 2$ ...

Hence result.

If i understand you correctly :from $a^2>b$ you get $16a^4>16b^2$ ,by raising 1st the inequality to the power of 2 and then multplying by 16.

But when the inequality is raised to 2 must we not have that b>0 ,otherwise the inequality may change if b<0.The problem assumes only $b\neq 0$ .

Example:take b=-3 and a=1 :we have $b<a^2$ ,but $b^2>a^4$

**Deadstar** · June 6th 2010, 06:05 PM

Originally Posted by alexandros

If i understand you correctly :from $a^2>b$ you get $16a^4>16b^2$ ,by raising 1st the inequality to the power of 2 and then multplying by 16.

But when the inequality is raised to 2 must we not have that b>0 ,otherwise the inequality may change if b<0.The problem assumes only $b\neq 0$ .

Example:take b=-3 and a=1 :we have $b<a^2$ ,but $b^2>a^4$

Yeah you're right.

Split it up into two cases then.

$a^2 > b$ we know.

So let case 1 be

$|a| > \sqrt{|b|}$ which was my first post and hence $a^4 > b^2$ .

Case 2 is

$|a| < \sqrt{|b|}$ and hence $a^4 < b^2$ .

From which we get that...

$16a^4 + a^2 b^2 - 16a^2b + 2ab + 4b^2 + 3 = (ab + 1)^2 + 2 + 16a^4 - 16a^2b + 4b^2$

$> (ab+1)^2 + 2 + 16a^4 - 16b^2 + 4b^2$

$> (ab+1)^2 + 2 + 16a^4 - 16a^4 + 4b^2 \geq 2$

Although I haven't checked I think some of the the > can be replaced with $\geq$ for one of the cases to include $a^2 > b$ with $a^4 = b^2$ . (This will only happen when b is negative since if b is +ve then $a^2 - b = 0$ )

**alexandros** · June 7th 2010, 03:28 AM

Originally Posted by Deadstar

Yeah you're right.

Split it up into two cases then.

$a^2 > b$ we know.

So let case 1 be

$|a| > \sqrt{|b|}$ which was my first post and hence $a^4 > b^2$ .

Case 2 is

$|a| < \sqrt{|b|}$ and hence $a^4 < b^2$ .

From which we get that...

$16a^4 + a^2 b^2 - 16a^2b + 2ab + 4b^2 + 3 = (ab + 1)^2 + 2 + 16a^4 - 16a^2b + 4b^2$

$> (ab+1)^2 + 2 + 16a^4 - 16b^2 + 4b^2$

$> (ab+1)^2 + 2 + 16a^4 - 16a^4 + 4b^2 \geq 2$

Although I haven't checked I think some of the the > can be replaced with $\geq$ for one of the cases to include $a^2 > b$ with $a^4 = b^2$ . (This will only happen when b is negative since if b is +ve then $a^2 - b = 0$ )

Forming the inequality : $16a^4 + a^2 b^2 - 16a^2b + 2ab + 4b^2 + 3 = (ab + 1)^2 + 2 + 16a^4 - 16a^2b + 4b^2$

$> (ab+1)^2 + 2 + 16a^4 - 16b^2 + 4b^2$

Do you assume : $-16a^2b>-16b^2$ ?

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