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If the roots of the equation bx^{2}+cx+a=0 be imaginary, then for all real values of x, the expression 3b^{2}x^{2}+6bcx+2c^{2} is

  • Option 1)

    Greater than 4ab

  • Option 2)

    Less than 4ab

  • Option 3)

    Greater than -4ab

  • Option 4)

    Less than -4ab

 

Answers (1)

As we learnt in 

Condition for Real and distinct roots of Quadratic Equation -

D= b^{2}-4ac> 0

- wherein

ax^{2}+bx+c= 0

is the quadratic equation

 

 and

Complex Roots with non - zero Imaginary part -

D= b^{2}-4ac< 0

- wherein

ax^{2}+bx+c= 0

is the quadratic equation

 

 bx^{2}+cx+a=0

if roots are imaginary , then

c^{2}-4ab< 0--------(1)

Expression is 3b^{2}x^{2}+6bcx+2c^{2}=y

Thus 3b^{2}x^{2}+6bcx+\left ( 2c^{2}-y \right )=0

For real value of x, we have

\left ( 6bc\right )^{2}-4\left ( 3b^{2} \right )\left (2c^{2}-y \right )\geq 0\\*\\*3c^{2}-2c^{2}+y\geq 0\\*\\*c^{2}+y\geq 0\\*\\*= > y\geq -c^{2}------(2)

Comparing (1) & (2)

y> -4ab


Option 1)

Greater than 4ab

Incorrect

Option 2)

Less than 4ab

Incorrect

Option 3)

Greater than -4ab

Correct

Option 4)

Less than -4ab

Incorrect

Posted by

Vakul

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