Key to "What is the condition that the following equation..."

Alan Zavala

Answered question

2022-04-16

What is the condition that the following equation has four real roots?

(a_{1} x^{2} + b_{1} x + c_{1}) (a_{2} x^{2} + b_{2} x + c_{2}) = 0

c_{1} c_{2} > 0

a_{1} c_{2} < 0

a_{2} c_{1} < 0

davelucasnp0j

Beginner2022-04-17Added 16 answers

It is sufficient if

a_{1} c_{1} < 0

and

a_{2} c_{2} < 0

.
If all 3 conditions hold then, multiplying the first and second

a_{1} c_{1} c_{2}^{2} < 0 so a_{1} c_{1} < 0

.
Similarly, multiplying the first and third,

a_{2} c_{2} c_{1}^{2} < 0

a_{2} c_{2} < 0

.
Therefore if all three conditions hold there are four real roots. The roots may not all be different depending on the coefficients.

Do you have a similar question?

Recalculate according to your conditions!

Didn't find what you were looking for?