Question

if the root of the equation( a2+ b2) x2+(a+c)2bx + b2+c2=0 is equal to zero are real and equal prove that a b c are in GP

Answer 1

$( \: {a}^{2} + {b}^{2} \: ) {x}^{2} - 2b( \: a + c \: )x \: + \: ( \: {b}^{2} + {c}^{2} \: ) = 0 \\ \\ Roots \: \: are \: \: real \: \: and \: \: equal \: , \\ therefore \: \: D=0 \\ \\ D = \: (- 2b( \: a + c \: )) {}^{2} - 4( \: {a}^{2} + {b}^{2} \: )( \: {b}^{2} + {c}^{2} \: ) = 0 \\ \\ {b}^{2} \: ( \: a {}^{2} + 2ac \: + {c }^{2} \: ) \: = {a}^{2} {b}^{2} + {a}^{2} {c}^{2} + {b}^{2} {c}^{2} + {b}^{4} \\ \\ ( \: ac \: ) {}^{2} - 2ac( {b}^{2} ) + ({b}^{2} ) {}^{2} = 0 \\ \\ \: \: \: ( \: ac \: - \: {b}^{2} \: ) {}^{2} \: = \: 0 \\ \\ \: \: \: \: \: \: \: \: \: \: \: \large \boxed { \: b {}^{2} \: = \: ac } \\ \\ a \: \: \: , \: \: \: b \: \: , \: \: c \: \: \: are \: \: in \: \: \: G.P.$