Sr Examen
Descomponer -y^4+5*y^2-12 al cuadrado
Solución
Expresión del cuadrado perfecto
Expresemos el cuadrado perfecto del trinomio cuadrático
$$\left(- y^{4} + 5 y^{2}\right) - 12$$
Para eso usemos la fórmula
$$a y^{4} + b y^{2} + c = a \left(m + y^{2}\right)^{2} + n$$
donde
$$m = \frac{b}{2 a}$$
$$n = \frac{4 a c - b^{2}}{4 a}$$
En nuestro caso
$$a = -1$$
$$b = 5$$
$$c = -12$$
Entonces
$$m = - \frac{5}{2}$$
$$n = - \frac{23}{4}$$
Pues,
$$- \left(y^{2} - \frac{5}{2}\right)^{2} - \frac{23}{4}$$
$$\left(- y^{4} + 5 y^{2}\right) - 12$$
Para eso usemos la fórmula
$$a y^{4} + b y^{2} + c = a \left(m + y^{2}\right)^{2} + n$$
donde
$$m = \frac{b}{2 a}$$
$$n = \frac{4 a c - b^{2}}{4 a}$$
En nuestro caso
$$a = -1$$
$$b = 5$$
$$c = -12$$
Entonces
$$m = - \frac{5}{2}$$
$$n = - \frac{23}{4}$$
Pues,
$$- \left(y^{2} - \frac{5}{2}\right)^{2} - \frac{23}{4}$$
Factorización
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$$\left(x + \left(\sqrt{2} \sqrt[4]{3} \cos{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{23}}{5} \right)}}{2} \right)} - \sqrt{2} \sqrt[4]{3} i \sin{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{23}}{5} \right)}}{2} \right)}\right)\right) \left(x + \left(\sqrt{2} \sqrt[4]{3} \cos{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{23}}{5} \right)}}{2} \right)} + \sqrt{2} \sqrt[4]{3} i \sin{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{23}}{5} \right)}}{2} \right)}\right)\right) \left(x + \left(- \sqrt{2} \sqrt[4]{3} \cos{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{23}}{5} \right)}}{2} \right)} + \sqrt{2} \sqrt[4]{3} i \sin{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{23}}{5} \right)}}{2} \right)}\right)\right) \left(x + \left(- \sqrt{2} \sqrt[4]{3} \cos{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{23}}{5} \right)}}{2} \right)} - \sqrt{2} \sqrt[4]{3} i \sin{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{23}}{5} \right)}}{2} \right)}\right)\right)$$
(((x + sqrt(2)*3^(1/4)*cos(atan(sqrt(23)/5)/2) + i*sqrt(2)*3^(1/4)*sin(atan(sqrt(23)/5)/2))*(x + sqrt(2)*3^(1/4)*cos(atan(sqrt(23)/5)/2) - i*sqrt(2)*3^(1/4)*sin(atan(sqrt(23)/5)/2)))*(x - sqrt(2)*3^(1/4)*cos(atan(sqrt(23)/5)/2) + i*sqrt(2)*3^(1/4)*sin(atan(sqrt(23)/5)/2)))*(x - sqrt(2)*3^(1/4)*cos(atan(sqrt(23)/5)/2) - i*sqrt(2)*3^(1/4)*sin(atan(sqrt(23)/5)/2))
Unión de expresiones racionales
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2 / 2\ -12 + y *\5 - y /
$$y^{2} \left(5 - y^{2}\right) - 12$$
-12 + y^2*(5 - y^2)