Play With Roots Of Equation

Calculus Level 3

If $a+b+c=0$ , then which of the following must be true regarding the quadratic equation $3ax^2 +2bx+c=0$ ?

It has one root in

(2,3)

and other in

(-2,-1)

It has at least one root in the interval

(0,1)

It has imaginary roots Can't be determined None of the above

1 solution

Rishi Sharma
May 17, 2016

Relevant wiki: Rolle's Theorem

$\\ Consider\quad f\left( x \right) =a{ x }^{ 3 }+b{ x }^{ 2 }+cx,\quad As\quad this\quad is\quad a\quad polynomial\quad function\\ it\quad is\quad continous\quad and\quad differentiable\quad in\quad \Re .\quad (\Re \quad is\quad the\quad set\quad of\quad all\quad real\quad numbers)\\ Now\quad we\quad notice\quad that\quad f\left( 0 \right) =f\left( 1 \right) =0\quad (\because \quad a+b+c=0)\\ So\quad by\quad applying\quad \quad Rolle's\quad Theorem\quad in\quad the\quad interval\quad (0,1)\quad we\quad can\quad say\quad that\\ \frac { d }{ dx } (f\left( x \right) )\quad will\quad have\quad atleast\quad one\quad zero\quad in\quad (0,1)\\ or\quad 3a{ x }^{ 2 }+2bx+c\quad will\quad have\quad atleast\quad one\quad zero\quad in\quad (0,1)$

interesting, but can you find the root ???

Frank Giordano - 4 years, 9 months ago

if we let b=-2c, then a=c and we have, assuming c not equal to zero, the factored form c(3x-1)(x-1) which has root x=1/3 inside the interval (0,1).

Frank Giordano - 4 years, 9 months ago

i used G-filtered Polycules to POINT to the solution, FYI https://www.facebook.com/groups/factorthis/

Frank Giordano - 4 years, 9 months ago

Play With Roots Of Equation

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