Two quadratic equations are given below: E_ 1 : ;3x^ 2 +(m-2)x+4=0, E_ 2 : ;x^ 2 +2mx+9=0

MATHEMATICS · Multi-part · AoPS · Mathematics · Quadratic Equations - Part 2 · Quadratic Equations - Part 2

Problem

Two quadratic equations are given below:

E_{1}:\;3x^{2}+(m-2)x+4=0,\qquad E_{2}:\;x^{2}+2mx+9=0,

where

m

is a real parameter. (a) Obtain the condition on

m

for which

E_{1}

possesses real roots. (b) Obtain the condition on

m

for which

E_{2}

possesses real roots. (c) Hence, find all real values of

m

for which both

E_{1}

and

E_{2}

have real roots simultaneously. (d) Determine the set of real values of

m

for which exactly one of the two equations has real roots.