xbox360

The problem is:
a,b,c are edges of triangle,
prove that the equation: ax^2+2(b+c)x+a=0 . has two distinct real root
​and my answer is:
a,b,c are edges therefore they are absolute values.
so
a>0
b>0
c>0

also
Δ=4(b+c)^2-4a^2
and I need to prove that Δ>0
so
4(b+c)^2-4a^2>0
which led me for:
(b+c)^2>a^2
b+c>a
now if so far i'm correct, than all I've to do is say that because of triangle inequality "b+c>a" is correct.
And I done with the proving.

Plato

MHF Helper
The problem is: a,b,c are edges of triangle,
prove that the equation: ax^2+2(b+c)x+a=0 . has two distinct real root.
You really need to work on terminology.
Triangles have sides not edges. You also mean that $a,~b,~\&~c$ are the lengths of the sides. That is the only that they are positive real numbers.

Moreover, you start by considering $ax^2+2(b+c)x+a=0$. If its discriminate, $\Delta\ne 0$ then that quadratic has two real roots.

What happens if $\Delta = 0~?$ that is proof by contradiction.

xbox360

You really need to work on terminology.
Triangles have sides not edges. You also mean that $a,~b,~\&~c$ are the lengths of the sides. That is the only that they are positive real numbers.

Moreover, you start by considering $ax^2+2(b+c)x+a=0$. If its discriminate, $\Delta\ne 0$ then that quadratic has two real roots.

What happens if $\Delta = 0~?$ that is proof by contradiction.
First of all, thank you for the quick replay. It is my mistake, I didn't mention that English was not my first language, so basically I do understand the terminology but not how to translate it properly
I might don't translate it accurately but in this problem, I've asked to proof that the quadratic has two real roots. Therefore, the only way I know is to proof that is by:
1. a $$\displaystyle \ne$$ 0 , which I know because it is part of the triangle
2. $$\displaystyle \Delta$$> 0 That I've proven as seen above. My question basically is, if my proof for that condition are correct.

Plato

MHF Helper
1. a $$\displaystyle \ne$$ 0 , which I know because it is part of the triangle
2. $$\displaystyle \Delta$$> 0 That I've proven as seen above. My question basically is, if my proof for that condition are correct.
Δ=4(b+c)^2-4a^2
and I need to prove that Δ>0
so
4(b+c)^2-4a^2>0
which led me for:
(b+c)^2>a^2
b+c>a
now if so far i'm correct, than all I've to do is say that because of triangle inequality "b+c>a" is correct.
Frankly, I would not give you much credit for that so-called proof.
Agreed: you do want to prove that $\Delta>0$ but it cannot be done by assuming that it is already true.

xbox360

Well I got stuck (Crying), I can't find a solution to proof that $$\displaystyle \Delta \neq 0$$ while $$\displaystyle a,b,c>0$$
Any help here please.

Last edited:

skeeter

MHF Helper
$\Delta = [2(b+c)]^2 - 4a^2$

$\Delta = 4(b+c)^2 - 4a^2$

$\Delta = 4[(b+c)^2 - a^2]$

$\Delta = 4(b+c+a)(b+c-a)$

$4 > 0$, $(b+c+a)>0$, and $(b+c)>a \implies (b+c-a)>0$

since all three factors are positive, $\Delta > 0 \implies ax^2+2(b+c)x+a=0$ has two distinct real roots.

1 person

xbox360

$$\displaystyle (b+c)>a \implies (b+c-a)>0$$
But if the parameters are:
$$\displaystyle a=3$$
$$\displaystyle b=2$$
$$\displaystyle c=1$$

then $$\displaystyle b+c-a=0$$
that led me into $$\displaystyle \Delta = 0$$ which I don't want.
Or am I missing something in your answer?

Plato

MHF Helper
But if the parameters are:
$$\displaystyle a=3$$ $$\displaystyle b=2$$ $$\displaystyle c=1$$
IMPOSSIBLE
Because then we would not have a triangle.

2 people