solve for $x$.

\x^4 + 125x^2 + 2500 = 0\

$x =$ (separate answers by a comma.)

if the answer is radical use sqrt(5) to denote $\sqrt{5}$ (use the correct radicand in the problem!)
if the answer is complex use i to denote $i$.

Question

solve for $x$.

\x^4 + 125x^2 + 2500 = 0\

$x =$   (separate answers by a comma.)

if the answer is radical use sqrt(5) to denote $\sqrt{5}$ (use the correct radicand in the problem!)
if the answer is complex use i to denote $i$.

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "blank_filling_formatter", "explain_then_answer", "tutor", "mastery_adaptive_brevity" ], "needs_drawing": false, "concepts_used": [ "Equations in Quadratic Form", "Quadratic Formula", "Simplifying Radicals", "Imaginary Unit" ], "new_concepts": [], "current_concepts": [ "Equations in Quadratic Form", "Quadratic Formula", "Simplifying Radicals", "Imaginary Unit" ] } ## Substitute to form a quadratic equation Using the *Equations in Quadratic Form* knowledge point $$ \begin{aligned} &\text{Let } u = x^2 \quad (u \ge 0 \text{ for real } x)\ &u^2 + 125u + 2500 = 0 \end{aligned} $$ ## Solve the quadratic equation for u Using the *Quadratic Formula* knowledge point $$ \begin{aligned} &u = \frac{-125 \pm \sqrt{125^2 - 4(1)(2500)}}{2(1)}\ &u = \frac{-125 \pm \sqrt{15625 - 10000}}{2}\ &u = \frac{-125 \pm \sqrt{5625}}{2}\ &u = \frac{-125 \pm 75}{2}\ &u_1 = \frac{-125 + 75}{2} = -25\ &u_2 = \frac{-125 - 75}{2} = -100 \end{aligned} $$ ## Solve for x using the imaginary unit Using the *Imaginary Unit* and *Simplifying Radicals* knowledge points $$ \begin{aligned} &x^2 = -25 \implies x = \pm \sqrt{-25} = \pm 5i\ &x^2 = -100 \implies x = \pm \sqrt{-100} = \pm 10i \end{aligned} $$ Solve for $x$. $$x^4 + 125x^2 + 2500 = 0$$ $x =$ $5i, -5i, 10i, -10i$ { "subject": "Mathematics", "question_type": "Fill-in-the-blank", "knowledge_point": [ "Mathematics", "Algebra", "Equations in Quadratic Form" ] }

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QUESTION IMAGE

Question

Explanation:

Substitute to form a quadratic equation

Solve the quadratic equation for u

Solve for x using the imaginary unit

Answer: