(i) Express the complex function f(x)=3 x^2+(1+2 i) x+2(i-1)...

(i) Express the complex function $f(x)=3 x^{2}+(1+2 i) x+2(i-1)$ in the form $f(x)=g(x)+i h(x)$, where $g(x)$ and $h(x)$ are real. (ii) solve $g(x)=0, h(x)=0$, then $f(x)=0 .$ (iii) find $|f(x)|^{2}$.

Key Concepts

Separation of Complex Functions into Real and Imaginary Parts

This concept involves expressing a complex function as the sum of a real function and an imaginary function multiplied by i. It is fundamental in complex analysis and algebra because it allows one to handle complex expressions by analyzing their real and imaginary components separately. For example, in any complex polynomial, coefficients involving i are split such that the expression can be written in the form g(x) + i h(x), where g(x) and h(x) are real-valued functions.

Solving Equations by Equating Real and Imaginary Parts

When a complex equation is set to zero, the condition of equality requires that both the real part and the imaginary part be independently equal to zero. This approach is used to find the roots of complex functions by solving two simultaneous equations: one for the real part and one for the imaginary part. It is essential in problems where one needs to determine the zeros of a complex function or to ensure the function vanishes.

Modulus (Absolute Value) of a Complex Function

The modulus of a complex number, often represented as |z|, is a measure of its magnitude, calculated as the square root of the sum of the squares of its real and imaginary parts. In many contexts, it is more practical to work with the squared modulus, |z|^2, which simplifies to the sum of the squares of the real and imaginary components. This concept is crucial in evaluating the size or norm of a complex function, particularly in applications involving distances or optimization in the complex plane.

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