// Numbas version: exam_results_page_options {"name": "Ecuaci\u00f3n de cuarto grado reducible a Cuadr\u00e1tica.jk ", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"statement": "

Resolver el siguiente ecuación:

\\[\\begin{eqnarray*}\\simplify[std]{x^4-{a^2+b^2}x^2+{(a*b)^2}}&=&{0}\\end{eqnarray*}\\]

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Observación: Para ingresar su respuesta, hágalo en el siguiente orden:

$x_1<x_2<x_3<x_4$

$x_1=$ [[0]] ; $x_2=$ [[1]] ; $x_3=$[[2]] ; $x_4=$[[3]]

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Input your answer as a fraction and not a decimal.

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Input your answer as a fraction and not a decimal.

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Input your answer as a fraction and not a decimal.

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Input your answer as a fraction and not a decimal.

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Utilicemos la sustitución $\\,x^2=t\\,$ en la ecuación:

\\[\\begin{eqnarray*}\\simplify[std]{x^4-{a^2+b^2}x^2+{(a*b)^2}}&=&{0}\\end{eqnarray*}\\]

Nos queda:

\\[\\begin{eqnarray*}\\simplify[std]{t^2-{a^2+b^2}t+{(a*b)^2}}&=&{0}\\end{eqnarray*}\\]

Esta ecuación la podemos resolver factorizando.

\\begin{align}
(\\simplify{t-{a^2}})(\\simplify{t-{b^2}})&=\\,{0}
\\end{align}

\\[\\simplify{(t-{a^2}={0})} \\,\\,\\,\\,;\\,\\,\\, \\simplify{(t-{b^2}={0})} \\]

\\[\\simplify{(t={a^2})} \\,\\,\\,\\,;\\,\\,\\, \\simplify{(t={b^2})} \\]

Ahora nos devolvemos con la sustitución $\\,x^2=t\\,$, de modo de encontrar los valores de la variable $x$:

\\[\\simplify{(x^2={a^2})} \\,\\,\\,\\,;\\,\\,\\, \\simplify{(x^2={b^2})} \\]

Estas ecuaciones las podemos resolver nuevamente factorizando:

\\[\\simplify{(x^2-{a^2}={0})} \\,\\,\\,\\,;\\,\\,\\, \\simplify{(x^2-{b^2}={0})} \\]

\\begin{align}
(\\simplify{x-{a}})(\\simplify{x+{a}})&=\\,{0}\\,\\,\\,\\,;\\,\\,\\,(\\simplify{x-{b}})(\\simplify{x+{b}})=\\,{0}
\\end{align}

\\begin{align}
\\simplify{(x={a})} \\,\\,\\,\\,;\\,\\,\\, \\simplify{x={-a}}\\,\\,\\,\\,;\\,\\,\\,\\simplify{(x={b})} \\,\\,\\,\\,;\\,\\,\\, \\simplify{x={-b}}
\\end{align}

Finalmente ordenamos las soluciones de menor a mayor como lo pide el enunciado:

\\begin{align}
\\simplify{(x_1={-b})} \\,\\,\\,\\,;\\,\\,\\, \\simplify{x_2={-a}}\\,\\,\\,\\,;\\,\\,\\,\\simplify{(x_3={a})} \\,\\,\\,\\,;\\,\\,\\, \\simplify{x_4={b}}
\\end{align}

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Shows how to define variables to stop degenerate examples.

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