Question

Show that there is no real number K belonging to [0,1] for which the equation $x^2-3x+k=0$ has two imaginary roots .

Answer 1

$x^2-3x+k=0$

For imaginary roots

$D<0$

$\Rightarrow (-3{)}^2-4(1)k<0$

$\Rightarrow 9-4k<0$

$\Rightarrow 4k-9>0$

$\Rightarrow k>\frac{9}{4}$

Imaginary roots occur only for $k\in (\frac{9}{4},\infty )$

$\therefore$ No imaginary roots when $k\in (0,1)$.