Question 549684
The discriminant for {{{2x^2+3kx-9=0}}} is
{{{(3k)^2-4*2*(-9)=9k^2+72}}}
The equation {{{2x^2+3kx-9=0}}} will have real and unequal roots when
{{{9k^2+72>0}}} which happens for all real values of k.
The discriminant for {{{x^2+(k-2)x+4=0}}} is
{{{(k-2)^2-4*1*4=(k-2)^2-16}}}
The equation {{{x^2+(k-2)x+4=0}}} will have equal roots when
{{{(k-2)^2-16=0}}} <--> {{{(k-2)^2=16}}} which happens for {{{k=6}}} and {{{k=-2}}}
Was the third equation {{{kx^2+4sqrt(3)x+k=0}}}?
The discriminant for {{{kx^2+4sqrt(3)x+k=0}}} is
{{{(4sqrt(3))^2-4*k*k=48-4k^2=4(12-k^2)}}}
The equation {{{kx^2+4sqrt(3)x+k=0}}} will have imaginary roots when
{{{12-k^2<0}}}, which happens for {{{abs(k)>sqrt(12)=2sqrt(3)}}}