Question

How to find the common roots of two cubic equations ? Also what is the logic behind it ?

Answer 1

hint:use the sum of roots thing

Answer 2

You can also directly factorize the cubics to find the common roots

Answer 3

Suppose the equations are x cube-7 xsquare+6x+1= 0 and x cube-6x square+x+7=0 ? How do we do this ?

Answer 4

Apologies for word notations

Answer 5

I'm not sure what you mean by 'common roots'. Are these roots of intersections between the two equations, or are these roots that both cubics share?

Answer 6

I'm assuming these are roots when the two equations meet (i.e solve simultaneously)

If you do subtract the two cubics i.e. do first cubic - second cubic, you should get
-x^2 + 5x - 6 = -(x^2 - 5x + 6) = -(x-3)(x-2).
Solving for zero, this means the two cubics meet when x = 3 or x =2

Answer 7

In the solution they have equated the two equations and found the values Like you did. Now they substituted the values of 3 and 2 in the two equations individually and found that it doesn't satisfy the given two equations. Then they concluded that it has no common roots so I didn't quite understand the logic behind why they did that

Answer 8

Ah I understand what they are doing. Two cubics (or any two equations in general) have roots whenever they meet. However there zeroes need not be the same. If the meeting point of the cubics produces x1 = x2 (i.e locations where the graphs meet) AND f(x1) = f(x2) then we say they have common ZEROES.

I'll use the word zero because that is less ambiguous than the term root. I'll illustrate below

Answer 9

A general example.