Question

Determine the nature of the roots of
${x}^{2} - 2bx + {b}^{2} + {c}^{2} = 0$

Answer 1

for determining nature of the roots you have to find the dicriminant of this quadratic
$d = {b}^{2} - 4ac \\ 4 {b}^{2} - 4 \times ( {b}^{2} + {c}^{2} ) \\ 4 {b}^{2} - 4 {b}^{2} - 4 {c}^{2} \\ = - 4 {c}^{2} \\ now \: {c }^{2} will \: be \: positive \: as \: all \: squares \: are \: positive \\ hence \: d < 0 \\ so \: the \: roots \: will \: be \: imaginary$