which values of $x$ and $y$ would make the following expression represent a real number?

$(4 + 5i)(x + yi)$

$x = 4, y = 5$
$x = -4, y = 0$
$x = 4, y = -5$
$x = 0, y = 5$

Question

which values of $x$ and $y$ would make the following expression represent a real number?

$(4 + 5i)(x + yi)$

$x = 4, y = 5$
$x = -4, y = 0$
$x = 4, y = -5$
$x = 0, y = 5$

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "multiple_choice_formatter", "explain_then_answer", "quick" ], "needs_drawing": false, "concepts_used": [ "Complex Number Multiplication" ], "new_concepts": [], "current_concepts": [ "Complex Number Multiplication", "Complex Number Simplification", "Real Numbers", "Imaginary Numbers" ] } ## Expand the complex product $$ \begin{aligned} (4 + 5i)(x + yi) &= 4x + 4yi + 5xi + 5yi^2 \ &= 4x + (4y + 5x)i - 5y \ &= (4x - 5y) + (4y + 5x)i \end{aligned} $$ ## Set the imaginary part to zero $$ 4y + 5x = 0 $$ ## Test the given options $$ \begin{aligned} &\text{Option 1: } x = 4, y = 5 \implies 4(5) + 5(4) = 40 e 0 \ &\text{Option 2: } x = -4, y = 0 \implies 4(0) + 5(-4) = -20 e 0 \ &\text{Option 3: } x = 4, y = -5 \implies 4(-5) + 5(4) = 0 \ &\text{Option 4: } x = 0, y = 5 \implies 4(5) + 5(0) = 20 e 0 \end{aligned} $$

x = 4, y = 5

x = -4, y = 0

x = 4, y = -5

x = 0, y = 5

{ "subject": "Mathematics", "question_type": "Multiple Choice", "knowledge_point": [ "Mathematics", "Algebra", "Complex Number Multiplication" ] }

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QUESTION IMAGE

Question

Explanation:

Expand the complex product

Set the imaginary part to zero

Test the given options

Answer: