Show that these have a common root

Asked Jun 02 '21 at 18:27

Active Jun 02 '21 at 20:53

Viewed 82 times

Show that these

$ax^2 + (b + d)x + c = 0$

$bx^2 + (c + d)x + a = 0$

$cx^2 + (a + d)x + b = 0$

have a common root if and only if $a+b+c+d = 0$ $a<b<c<d$ and non-zero

It is quite clear that when $x=1$, the root is confirmed. However, adding all equations is not giving something useful. Is there some method to solve such questions?

Moreover, can a solution using determinants be possible?

Any hints appreciated.

edited Jun 02 '21 at 20:53

asked Jun 02 '21 at 18:27

mathx

The "if-and-only-if" statement means that whatever proof is provided, it needs to work both ways – imranfat Jun 02 '21 at 18:30
The question still remains... – mathx Jun 02 '21 at 18:32
3

On AoPS: https://artofproblemsolving.com/community/c6h421070p2378205 – Martin R Jun 02 '21 at 18:32
@MartinR Perfect – mathx Jun 02 '21 at 18:40
@mathx The "only if" part requires the additional assumption that $a,b,c$ are not all equal. – dxiv Jun 02 '21 at 19:34
@dxiv Let $a>b>c>d$ – mathx Jun 02 '21 at 19:58
@mathx You should edit that into the question, not as a comment. The condition is stronger than needed, the only other solution is $a=b=c$ (when the three equations become identical). – dxiv Jun 02 '21 at 20:36

Show that these have a common root

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