Sr Examen
Descomponer -y^2+y*x+8*x^2 al cuadrado
Solución
Factorización
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/ / ____\\ / / ____\\ | y*\-1 + \/ 33 /| | y*\1 + \/ 33 /| |x - ---------------|*|x + --------------| \ 16 / \ 16 /
$$\left(x - \frac{y \left(-1 + \sqrt{33}\right)}{16}\right) \left(x + \frac{y \left(1 + \sqrt{33}\right)}{16}\right)$$
(x - y*(-1 + sqrt(33))/16)*(x + y*(1 + sqrt(33))/16)
Expresión del cuadrado perfecto
Expresemos el cuadrado perfecto del trinomio cuadrático
$$8 x^{2} + \left(x y - y^{2}\right)$$
Escribamos tal identidad
$$8 x^{2} + \left(x y - y^{2}\right) = - \frac{33 y^{2}}{32} + \left(8 x^{2} + x y + \frac{y^{2}}{32}\right)$$
o
$$8 x^{2} + \left(x y - y^{2}\right) = - \frac{33 y^{2}}{32} + \left(2 \sqrt{2} x + \frac{\sqrt{2} y}{8}\right)^{2}$$
en forma de un producto
$$\left(- \sqrt{\frac{33}{32}} y + \left(2 \sqrt{2} x + \frac{\sqrt{2}}{8} y\right)\right) \left(\sqrt{\frac{33}{32}} y + \left(2 \sqrt{2} x + \frac{\sqrt{2}}{8} y\right)\right)$$
$$\left(- \frac{\sqrt{66}}{8} y + \left(2 \sqrt{2} x + \frac{\sqrt{2}}{8} y\right)\right) \left(\frac{\sqrt{66}}{8} y + \left(2 \sqrt{2} x + \frac{\sqrt{2}}{8} y\right)\right)$$
$$\left(2 \sqrt{2} x + y \left(\frac{\sqrt{2}}{8} + \frac{\sqrt{66}}{8}\right)\right) \left(2 \sqrt{2} x + y \left(- \frac{\sqrt{66}}{8} + \frac{\sqrt{2}}{8}\right)\right)$$
$$\left(2 \sqrt{2} x + y \left(\frac{\sqrt{2}}{8} + \frac{\sqrt{66}}{8}\right)\right) \left(2 \sqrt{2} x + y \left(- \frac{\sqrt{66}}{8} + \frac{\sqrt{2}}{8}\right)\right)$$
$$8 x^{2} + \left(x y - y^{2}\right)$$
Escribamos tal identidad
$$8 x^{2} + \left(x y - y^{2}\right) = - \frac{33 y^{2}}{32} + \left(8 x^{2} + x y + \frac{y^{2}}{32}\right)$$
o
$$8 x^{2} + \left(x y - y^{2}\right) = - \frac{33 y^{2}}{32} + \left(2 \sqrt{2} x + \frac{\sqrt{2} y}{8}\right)^{2}$$
en forma de un producto
$$\left(- \sqrt{\frac{33}{32}} y + \left(2 \sqrt{2} x + \frac{\sqrt{2}}{8} y\right)\right) \left(\sqrt{\frac{33}{32}} y + \left(2 \sqrt{2} x + \frac{\sqrt{2}}{8} y\right)\right)$$
$$\left(- \frac{\sqrt{66}}{8} y + \left(2 \sqrt{2} x + \frac{\sqrt{2}}{8} y\right)\right) \left(\frac{\sqrt{66}}{8} y + \left(2 \sqrt{2} x + \frac{\sqrt{2}}{8} y\right)\right)$$
$$\left(2 \sqrt{2} x + y \left(\frac{\sqrt{2}}{8} + \frac{\sqrt{66}}{8}\right)\right) \left(2 \sqrt{2} x + y \left(- \frac{\sqrt{66}}{8} + \frac{\sqrt{2}}{8}\right)\right)$$
$$\left(2 \sqrt{2} x + y \left(\frac{\sqrt{2}}{8} + \frac{\sqrt{66}}{8}\right)\right) \left(2 \sqrt{2} x + y \left(- \frac{\sqrt{66}}{8} + \frac{\sqrt{2}}{8}\right)\right)$$
Unión de expresiones racionales
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2 8*x + y*(x - y)
$$8 x^{2} + y \left(x - y\right)$$
8*x^2 + y*(x - y)