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Algebra Level 3

Consider three reals $a,b,c$ satisfying $a+b+c=0$

Find the ratio of the roots of the equation $ax^2+bx+c=0$ .

This is an original problem and belongs to the set My creations

\dfrac{b}{a}

\dfrac{c}{b}

\dfrac{c}{a}

\dfrac{b}{c}

1 solution

Skanda Prasad
Sep 18, 2017

Given that $a+b+c=0$

For a quadratic equation $ax^2+bx+c=0$ , if $a+b+c=0$ , then the roots are $1$ and $\dfrac{c}{a}$ .

Hence the ratio is $\dfrac{c}{a}$ .

Note:- $\dfrac{a}{c}$ is also a ratio but intentionally I have not given it in the options.

If a+b+c=0 Then Adding 2 ab+2bc+2ca we get an+BC+ca=a²+b²+c²=0

Saksham Jain - 3 years, 8 months ago

How? Can u kindly use Latex?

Skanda Prasad - 3 years, 8 months ago

Please expand and show...I tried...I'm not getting that...

Skanda Prasad - 3 years, 8 months ago

I will try to use latex or upload photo .i think you edited the question. plzz tell me original question,i forgot

Saksham Jain - 3 years, 8 months ago

This was the original question:- $\dfrac{a^2}{bc}+\dfrac{b^2}{ac}+\dfrac{c^2}{ab}=3$

and $a^{2}+b^{2}+c^{2}\neq ab+bc+ac$

The same thing continued which is there presently in the question.

Skanda Prasad - 3 years, 8 months ago