BIOLOGY HIGH SCHOOL

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

Answers

Answer 1
Answer: 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
Answer:

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| = √(x^2 + y^2) \ }
  • Remember this \boxed{ \ i^2 = (√(-1))^2 = -1 \ }

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

  • Re(z) = 3
  • Im(z) = -4
  • The modulus \boxed{ \ |z| = √(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{ \ (x)/(y) = (-4)/(5) \ or (x)/(y) = (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 brainly.com/question/3269852
  2. The piecewise-defined functions brainly.com/question/9590016
  3. The composite function brainly.com/question/1691598

Keywords: identify, the values of x and y, the following expression, represent a real number, (4 + 5i)(x + yi), a complex number, real, imaginary

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