How to disprove a Fundamental Theory

Algebra Level 3

How many real roots does

( x − a ) ( x − b ) ( c − a ) ( c − b ) + ( x − b ) ( x − c ) ( a − b ) ( a − c ) + ( x − a ) ( x − c ) ( b − a ) ( b − c ) = 1 \frac{(x-a)(x-b)}{(c-a)(c-b)}+\frac{(x-b)(x-c)}{(a-b)(a-c)}+\frac{(x-a)(x-c)}{(b-a)(b-c)}=1

have with a ≠ b ≠ c ≠ 0 a\neq b\neq c\neq 0 ?

0 Infinite Roots 2 3

View wiki
Nanayaranaraknas Vahdam by Nanayaranaraknas Vahdam , 22, India

This section requires Javascript.
You are seeing this because something didn't load right. We suggest you, (a) try refreshing the page, (b) enabling javascript if it is disabled on your browser and, finally, (c) loading the non-javascript version of this page . We're sorry about the hassle.

1 solution

Easiest way to solve is by inspection. It is evident that the equation is at most quadratic. Thus, it should have at most 2 2 roots. However, plugging in x = a , b , c x=a,b,c , all three satisfy the equality. If there are 3 3 distinct roots, then there must be a relation between a , b , c a,b,c independent of x x . Thus, the equation is actually an identity, so there are i n f i n i t e \boxed{infinite} roots.

19 Helpful 0 Interesting 0 Brilliant 0 Confused

Thumbs up..!!

Deepanshu Gupta - 6 years, 8 months ago

That was awesome,,

Mvs Saketh - 6 years, 9 months ago

nice solution

Mardokay Mosazghi - 6 years, 8 months ago

Yep, a nice problem solving tool.

Daniel Liu - 6 years, 8 months ago

A nice logic. Congratulations.

Niranjan Khanderia - 6 years, 8 months ago

0 pending reports

×

Problem Loading...

Note Loading...

Set Loading...