Homework Help: Quadratic Equations - Condition for real roots

1. Jul 17, 2013

Saitama

1. The problem statement, all variables and given/known data
Let $a,b,c$ and $m \in R^{+}$. Find the range of $m$ (independent of $a,b$ and $c$) for which at least one of the following equations, $ax^2+bx+cm=0, bx^2+cx+am=0$ and $cx^2+ax+bm=0$ have real roots.

2. Relevant equations

3. The attempt at a solution
I don't really know where to start. If the quadratic has real roots, the discriminant is greater than or equal to zero. This would give me three inequations for each quadratic but how should I deal with "at least" one of the equation having real roots?

2. Jul 17, 2013

Mandelbroth

$$b^2-4acm \geq 0 \\ c^2-4bam \geq 0 \\ a^2-4cbm \geq 0$$
I'm confused as to your meaning in the first sentence. Was what you meant equivalent to $\{a,b,c,m\}\subset\mathbb{R}_{+}$?

3. Jul 17, 2013

Staff: Mentor

Break the problem into three cases.
1. Assume that the first equation has real roots.
2. Assume that the second equation has real roots.
3. Assume that the third equation has real roots.

For each case, what can you say about the discriminant, which will have a different form for each case? I haven't worked the problem, but this is how I would approach it.

4. Jul 17, 2013

HallsofIvy

Adding to what Mandelbroth said, $b^2- 4acm\ge 0$ is equivalent to $m\le b^2/4ac$, $c^2- 4bam\ge 0$ is equivalent to $m\le c^2/4ab$, and $a^2- 4cbm\ge 0$ is equivalent to $m\le a^2/4bc$. If you do not know any relations among a, b, and, c, the most you can say is that at least one of those must be true.

5. Jul 17, 2013

I like Serena

One of the 3 inequalities with $m$ on the left side must have the greatest value on the right hand side.
That would be the one with the greatest value in the numerator.

Suppose $a$ has the greatest value.
So $a \ge b$ and $a \ge c$.
Then what can you say about this greatest right hand value for sure?

6. Jul 17, 2013

haruspex

I suggest it would be easier to invert the logic: suppose none of them has real roots. Then you can look for a way of combining the inequalities that eliminates a, b and c. For an m that satisfies the condition that emerges, there likely exists a triple (a, b, c) s.t. none of the equations has real roots (but you'll need to show the triple does exist). Finally, you need to show that for all other m no such triple exists.

7. Jul 18, 2013

Saitama

I have three inequalities,
$m\le b^2/4ac, m\le c^2/4ab$ and $m\le a^2/4bc$.

For $a\geq b$ and $a \geq c$, $a^2/4bc$ has the greatest value. What next?

When none of them has real roots, the inequalities are
$m> b^2/4ac, m > c^2/4ab$ and $m > a^2/4bc$
Adding the inequalities,
$$12m>\frac{a^3+b^3+c^3}{abc}$$
What should I do next? I can find the minimum value of RHS of inequality but I don't think that would help. :uhh:

8. Jul 18, 2013

I like Serena

With $a\geq b$ and $a \geq c$, you get that $a^2 \ge bc$, ...

9. Jul 18, 2013

Saitama

The only thing I understand is that the minimum value of $a^2/(4bc)$ is 1/4.

10. Jul 18, 2013

verty

The point is, which m satisfies all inequalities? For m to be greater than all 3, it must be greater than the greatest.

Last edited: Jul 18, 2013
11. Jul 18, 2013

haruspex

Adding them didn't get rid of a, b and c. Try something else.... something, shall we say, more productive

12. Jul 19, 2013

verty

Oh, I didn't understand what was meant by "independent of a,b,c", I see now that it means "find a range of m that will guarantee one of these has real solutions given a,b,c > 0".

13. Jul 19, 2013

I like Serena

Yes. So without any knowledge of a, b, and c, you can still guarantee that there will be a solution as long as $0<m \le \,^1\!\!/\!_4$.

14. Jul 19, 2013

Saitama

Multiply the inequalities?
That gives $$m>\frac{1}{4}$$
I think this means that m should be less than or equal to 1/4 so that at least one of them have real roots. So I guess this is enough for the question?

Yep!

Both the solutions are interesting, thank you both ILS and haruspex!

15. Jul 19, 2013

haruspex

You've shown that no matter what a, b and c are, if there are no roots then m > 1/4.
$\forall a \in\Re^+ \forall b \in\Re^+ \forall c \in\Re^+, m \in [0\frac 14] \Rightarrow \exists (a root)$
Equivalently
$m \in [0\frac 14] \Rightarrow (\forall a \in\Re^+ \forall b \in\Re^+ \forall c \in\Re^+, \exists (a root))$
The question (I think) asks for the complete range R of m for which, no matter what a, b and c are, there are guaranteed some roots.
$m \in R \Leftrightarrow (\forall a \in\Re^+ \forall b \in\Re^+ \forall c \in\Re^+, \exists (a root))$

Last edited: Jul 19, 2013
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