dPalp1eec4hicksj
Answered

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

(4 + 5i)(x + yi)

Ax = 4, y = 5
Bx = –4, y = 0
Cx = 4, y = –5
Dx = 0, y = 5

Verified Answer :

taskmasters

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

BladeRunner212

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

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

Related Questions