Question

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

A. x = 4, y = 5
B. x = –4, y = 0
C. x = 4, y = –5
D. x = 0, y = 5

Answer 1

I think the correct answer from the choices listed above is option C. The values of x and y that would make the expression to represent a real number would be that  x = 4, y = –5. Substituting and simplifying,

(4 + 5i)(4 - 5i)
16 - 20i + 20i - 25i²
16 - 25(-1)
16 +25
41

Answer 2

x = 4, y = -5

Further explanation

This is a problem with complex numbers.

A complex number is a number of the form $\boxed{ \ z = x + yi \ }$ where x and y are both real  numbers.

• x is considered the real part of the complex  number.
• y is considered the imaginary part of the complex  number.
• The real part of a complex number $\boxed{ \ z = x + yi \ }$ is denoted by $\boxed{ \ Re(z) \ }.$
• The imaginary part of a complex number is denoted by $\boxed{ \ Im(z) \ }.$

• The modulus of a complex number $\boxed{ \ z = x + yi \ }$ is denoted by $\boxed{ \ |z| \ }$ and defined to  be $\boxed{ \ |z| = \sqrt{x^2 + y^2} \ }$
• Remember this $\boxed{ \ i^2 = (\sqrt{-1})^2 = -1 \ }$

For example, $\boxed{ \ z = 3 - 4i \ }$

• Re(z) = 3
• Im(z) = -4
• The modulus $\boxed{ \ |z| = \sqrt{3^2 + (-4)^2} = 5 \ }$  

Let's look at the key problem.

We must identify the values of x and y that would make the following expression represent a real number.

$\boxed{ \ = (4 + 5i)(x + yi) \ }$

$\boxed{ \ = 4(x + yi) + 5i(x + yi) \ }$

$\boxed{ \ = 4x + 4yi + 5xi + 5yi^2 \ }$

$\boxed{ \ = 4x + 4yi + 5xi - 5y \ }$

At this point, 4yi and 5xi must be of equal value and marked opposite, so that the expression represents real numbers.

$\boxed{ \ 4y = - 5x \ }$

Become a fraction, i.e.,

$\boxed{ \ \frac{x}{y} = \frac{-4}{5} \ or \frac{x}{y} = \frac{4}{-5} \ }$

From the options, the answer is C.

$\boxed{ \ x = 4, y = -5 \ }$

- - - - - - -

Let's check back to expression.

$\boxed{ \ = (4 + 5i)(x + yi) \ }$

$\boxed{ \ = 4x + 4yi + 5xi - 5y \ }$

x = 4 dan y = - 5 ⇒ $\boxed{ \ = 4(4) + 4(-5)i + 5(4)i - 5(-5) \ }$

$\boxed{ \ = 16 + -20i + 20i + 25 \ }$

$\boxed{ \ = 16 + 25 \ }$

$\boxed{ \ = 16 + 25 \ }$

Thus the expression represent a real number of 41.

Learn more

1. The midpoint https://brainly.com/question/3269852
2. The piecewise-defined functions https://brainly.com/question/9590016
3. The composite function https://brainly.com/question/1691598

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