find all zeros of $x^4 + x^2 - 72 = 0$. enter the zero separated by commas. enter exact value, not decimal approximations.

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{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "blank_filling_formatter", "explain_then_answer", "tutor", "mastery_adaptive_brevity" ], "needs_drawing": false, "concepts_used": [ "Factoring Quadratics", "Zero Product Property", "Square Root Property", "Imaginary Unit" ], "new_concepts": [ "Equations in Quadratic Form", "Quartic Equations" ], "current_concepts": [ "Factoring Quadratics", "Zero Product Property", "Square Root Property", "Imaginary Unit", "Equations in Quadratic Form", "Quartic Equations" ] } ## Substitute to form a quadratic equation Using the *Factoring Quadratics* knowledge point $$ \begin{aligned} &\text{Let } u = x^2 \text{ where } u \ge 0 \text{ for real } x.\ &u^2 + u - 72 = 0\ &(u + 9)(u - 8) = 0 \end{aligned} $$ ## Solve for the substituted variable Using the *Zero Product Property* knowledge point $$ \begin{aligned} &u + 9 = 0 \implies u = -9\ &u - 8 = 0 \implies u = 8 \end{aligned} $$ ## Solve for x using the square root property Using the *Square Root Property* and *Imaginary Unit* knowledge points $$ \begin{aligned} &x^2 = -9 \implies x = \pm \sqrt{-9} = \pm 3i\ &x^2 = 8 \implies x = \pm \sqrt{8} = \pm 2\sqrt{2} \end{aligned} $$ ## List all zeros The complete set of zeros consists of both real and complex solutions. $$ x = 3i, -3i, 2\sqrt{2}, -2\sqrt{2} $$ Find all zeros of $x^4 + x^2 - 72 = 0$. Enter the zero separated by commas. Enter exact value, not decimal approximations. $3i, -3i, 2\sqrt{2}, -2\sqrt{2}$ { "subject": "Mathematics", "question_type": "Fill-in-the-blank", "knowledge_point": [ "Mathematics", "Algebra", "Equations in Quadratic Form" ] }

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Substitute to form a quadratic equation

Solve for the substituted variable

Solve for x using the square root property

List all zeros

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