Question

If $\alpha, \beta \in R$ are such that 1$-$2i (here $i^2$=$-$1) is a root of z$^2$+$\alpha$z+$\beta$=0, then ($\alpha-\beta$) is equal to:

• A. 3
• B. -3
• C. 7
• D. -7

Hint:

Whenever a polynomial equation has real coefficients and a complex number $a+bi$ is a root, you can be certain that its conjugate $a-bi$ is also a root. This is a fundamental theorem that simplifies many problems involving complex roots.

Answer 1

The Correct Option is D

Since coefficients are real, the other root is \(1+2i\). Sum of roots: \[ (1-2i)+(1+2i)=2=-\alpha \Rightarrow \alpha=-2 \] Product of roots: \[ (1-2i)(1+2i)=1+4=5=\beta \] \[ \alpha-\beta=-2-5=-7 \] Correct option: (D)