Question

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

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 $k\in (0,1)$.