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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)

best_answer

 

Discriminant of Quadratic Equation -

D=b^{2}-4ac

- wherein

ax^{2}+bx+c= 0

is the quadratic equation

 

 

 

Complex Roots with non - zero Imaginary part -

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

- wherein

ax^{2}+bx+c= 0

is the quadratic equation

 

 

Given bx^{2}+cx+a=0 has imaginary roots

\Rightarrow c^{2}-4ab<0\; \; \; \; \; \Rightarrow c^{2}<4ab\; \; \; \; \; \Rightarrow -c^{2}>-4ab\; \; \; \; \; \; \; \; \; \; \; \; \; .....(i)

Letf(x)=3b^{2}x^{2}+6bcx+2c^{2}

Here, 3b^{2}>0

So, the given expression has a minimum value

\therefore\; \; minimum\; value=\frac{-D}{4a}

\frac{4(3b^{2})(2c^{2})-36b^{2}c^{2}}{4(3b^{2})}=-\frac{12b^{2}c^{2}}{12b^{2}}=-c^{2}>-4ab         [From eq.(i)]

Posted by

Divya Prakash Singh

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