Question

If ad # bc then prove that equation

$(a {}^{2} + b {}^{2} )x {}^{2} + 2(ac + bd)x + (c {}^{2} + d {}^{2} ) = 0$

has no real roots.


[CBSE 2017 ]

Answer 1

hey mate
here's the solution

Answer 2

$\huge \bf \pink {Hey \: there!! }$


The given quadratic equation is :-

→ ( a² + b² )x² + 2( ac + bd )x + ( c² + d² ) = 0.


Comparing it with Ax² + Bx + C = 0, we get

A = ( a² + b² ) , B = ( ac + bd ) and C = ( c² + d² ) .


°•° D = B² - 4AC .

= [ 2( ac + bd ) ]² - 4( a² + b² ) ( c² + d² )

= 4( ac + bd )² - 4[ (ac)² + (ad)² + (bc)² + (bd)² ] .

= 4[ (ac)² + (bd)² + 2acbd ] - 4(ac)² - 4(ad)² - 4(bc)² - 4(bd)² .

= 4(ac)² + 4(bd)² + 8acbd - 4(ac)² - 4(ad)² - 4(bc)² - 4(bd)² .

= - 4(ad)² - 4(bc)² + 8acbd .

= -4[ (ad)² + (bc)² - 2acbd ] .

= -4( ad - bc )² < 0 . [ When ad ≠ bc . ]


So, the given quadratic equation has no real roots , when ad ≠ bc .


✔✔ Hence, it is proved ✅ ✅.

____________________________________



$\huge \green{ \boxed{ \boxed{ \mathbb{THANKS}}}}$



$\huge \bf \orange{ \#BeBrainly.}$