Determining Real Numbers in the Expression (4 + 5i)(x + yi) | Explained with Complex Numbers Properties and Equations

Which values of x and y would make the following expression represent a real number?(4 + 5i)(x + yi)

To determine the values of x and y that would make the expression (4 + 5i)(x + yi) represent a real number, we need to consider the properties of complex numbers

To determine the values of x and y that would make the expression (4 + 5i)(x + yi) represent a real number, we need to consider the properties of complex numbers.

First, let’s simplify the expression by performing the multiplication:

(4 + 5i)(x + yi) = 4x + 4yi + 5ix + 5i^2

Remember that i^2 is equal to -1, so the expression becomes:

4x + 4yi + 5ix + 5(-1)
4x + 4yi + 5ix – 5

Now, in order to represent a real number, the imaginary part of the expression (4yi + 5ix) should be equal to zero. This means that the coefficient of y (4y) and the coefficient of i multiplied by x (5x) should both equal zero.

Therefore, we get two equations:

4y = 0 — (1)
5x = 0 — (2)

From equation (1), we can see that y must be equal to zero in order for the expression to be real.

From equation (2), x must also be equal to zero for the expression to be real.

Thus, the values of x and y that would make the expression (4 + 5i)(x + yi) represent a real number are x = 0 and y = 0.

In conclusion, when both x and y are equal to zero, the expression (4 + 5i)(x + yi) represents a real number.

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