select the correct answer.

the complex solutions of which equation have a real component of 4?

$x^2 + 4x + 4 = -25$
$x^2 - 4x + 4 = -25$
$x^2 + 8x + 16 = -21$
$x^2 - 8x + 16 = -21$

Question

select the correct answer.

the complex solutions of which equation have a real component of 4?

$x^2 + 4x + 4 = -25$
$x^2 - 4x + 4 = -25$
$x^2 + 8x + 16 = -21$
$x^2 - 8x + 16 = -21$

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "multiple_choice_formatter", "explain_then_answer", "quick" ], "needs_drawing": false, "concepts_used": [ "Perfect Square Trinomials", "Square Root Property", "Complex Roots" ], "new_concepts": [], "current_concepts": [ "Perfect Square Trinomials", "Square Root Property", "Complex Roots" ] } ## Identify the structure of the equations $$ \begin{aligned} &x^2 \pm 2dx + d^2 = (x \pm d)^2 \ &(x - d)^2 = -k \implies x = d \pm i\sqrt{k} \quad (k > 0) \ &\text{Real component} = d = 4 \end{aligned} $$ ## Match the perfect square trinomial with $d = 4$ $$ \begin{aligned} &(x - 4)^2 = x^2 - 8x + 16 \ &x^2 - 8x + 16 = -21 \end{aligned} $$ ## Solve the matched equation to verify $$ \begin{aligned} &(x - 4)^2 = -21 \ &x - 4 = \pm i\sqrt{21} \ &x = 4 \pm i\sqrt{21} \end{aligned} $$

(A) $x^2 + 4x + 4 = -25$

(B) $x^2 - 4x + 4 = -25$

(C) $x^2 + 8x + 16 = -21$

(D) $x^2 - 8x + 16 = -21$

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

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QUESTION IMAGE

Question

Explanation:

Identify the structure of the equations

Match the perfect square trinomial with \(d = 4\)

Solve the matched equation to verify

Answer: