6. PYQ - Zeroes of a quadratic polynomial

Evaluate the zeroes of the quadratic polynomial \(4x^2 - (5 + 4 \sqrt{3}) x + 5\sqrt{3}\) and verify the relationship between the zeroes and coefficients of the polynomial. 

 

Zeroes: ​$\frac{i}{i}$​ and ​$\sqrt{i}$

 

Verification:

 

​$\mathrm{\alpha }+\mathrm{\beta }=\sqrt{i}+\frac{i}{i}$

 

\(-\frac{\text{coefficient of }x}{\text{coefficient of }x^2}\)​$=\sqrt{i}+\frac{i}{i}$

 

​$\mathrm{\alpha }\mathrm{\beta }=\frac{i\sqrt{i}}{i}$

 

\(\frac{\text{constant}}{\text{coefficient of }x^2}\)​$=\frac{i\sqrt{i}}{i}$

 

Hence verified.