Grade 11Algebra

Let abc be sides of triangle . No two are equal and € belongs to real numbers.If roots of x.x +2(a+b+c)x+3€(ab+bc+ca)=0 are real. Then range of€=?

Angad Bajwa

9 Years agoGrade 11

2 Answers

Vikas TU

9 Years ago

For roots to be real its Discriminant should be greater than equal to zero.

It should not be imaginary.

Hence,

4(a+b+c)^2 – 12€(ab+bc+ca) greater than equal to zero.

mycroft holmes

9 Years ago

In triangles we have the inequality

$ab+bc+ca \le a^2+b^2+c^2 \le 2(ab+bc+ca)$ which can be written as

$3(ab+bc+ca) \le (a+b+c)^2 \le 4(ab+bc+ca)$

The left inequality arises from $(a-b)^2+(b-c)^2+(c-a)^2 \ge 0$ and the right inequality from the triangle inequalities squared i.e.

$(a-b)^2<c^2 \\ \ (b-c)^2 <a^2 \\ \ (c-a)^2 <b^2$

and summing up.

If $(a+b+c)^2 - 3 \epsilon (ab+bc+ca) \ge 0$ then the quadratic has real roots

So, we need $\epsilon \le 1$ if it has to be true for any value of a,b,c

Also, if $\epsilon \ge \frac{4}{3}$ then it surely has imaginary roots. For $1 \le \epsilon \le \frac{4}{3}$ it is ambiguous whether the roots are real or imaginary.

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Let abc be sides of triangle . No two are equal and € belongs to real numbers.If roots of x.x +2(a+b+c)x+3€(ab+bc+ca)=0 are real. Then range of€=?

Angad Bajwa

2 Answers

Vikas TU

mycroft holmes

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