Question

How do find the product of \[7 - 2i\] and its conjugate.

Answer 1

Hint: This problem comes under complex numbers. Complex numbers means algebraic expression in term imaginary numbers say $i$ which can be expressed in the form $a + ib$, where a and b are real numbers $i$ is imaginary number, ${i^2} = - 1$. In this problem we need to find the product of the conjugate of the given complex number. Then we find the solution with basic mathematical calculation.

Formula used: $(x + iy)(x - iy) = {x^2} - {i^2}{y^2} + ixy - ixy$
$(x + iy)(x - iy) = {x^2} + {y^2}$

Complete step-by-step solution:
Let us consider a given complex number $7 - 2i$
Now, the conjugate $7 - 2i$ is
In this the sign of the imaginary part of a given complex number is negative, so the conjugate will be the imaginary part will become negative and the real part remains the same.
$ \Rightarrow 7 + 2i$
Now the product of given complex number with conjugate is
\[ \Rightarrow (7 - 2i)(7 + 2i)\]
Now used formula mentioned in formula used, we get
$ \Rightarrow {7^2} - {2^2}{i^2}$
We know that ${i^2} = - 1$ by simplifying it in the above equation and finding square of given number, we get
$ \Rightarrow 49 + 4$
On adding the term and we get,
$ \Rightarrow 53$

Therefore, the product of given complex number and its conjugate is $53$

Note: Complex numbers appear when an imaginary part of the equation comes that is negative integers in root value. We need to know about real and imaginary parts of complex numbers.
Conjugate in complex numbers means that the change of signs in the imaginary part of the complex. If a positive sign in an imaginary part is a complex number then the conjugate of complex numbers in the imaginary part is negative and vice versa.
The complex number expressed as $a + ib$. Then to find the complex conjugate and with complex multiplication and properties we solve this. We need to be familiar with that so that it will be easy to solve.