Sr Examen
Descomponer -y^2-y*x+8*x^2 al cuadrado
Solución
Ha introducido
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2 2 - y - y*x + 8*x
$$8 x^{2} + \left(- x y - y^{2}\right)$$
-y^2 - y*x + 8*x^2
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
$$\left(- \sqrt{\frac{33}{32}} y + \left(2 \sqrt{2} x + - \frac{\sqrt{2}}{8} y\right)\right) \left(\sqrt{\frac{33}{32}} y + \left(2 \sqrt{2} x + - \frac{\sqrt{2}}{8} y\right)\right)$$
$$\left(- \frac{\sqrt{66}}{8} y + \left(2 \sqrt{2} x + - \frac{\sqrt{2}}{8} y\right)\right) \left(\frac{\sqrt{66}}{8} y + \left(2 \sqrt{2} x + - \frac{\sqrt{2}}{8} y\right)\right)$$
$$\left(2 \sqrt{2} x + y \left(- \frac{\sqrt{2}}{8} + \frac{\sqrt{66}}{8}\right)\right) \left(2 \sqrt{2} x + y \left(- \frac{\sqrt{66}}{8} - \frac{\sqrt{2}}{8}\right)\right)$$
$$\left(2 \sqrt{2} x + y \left(- \frac{\sqrt{2}}{8} + \frac{\sqrt{66}}{8}\right)\right) \left(2 \sqrt{2} x + y \left(- \frac{\sqrt{66}}{8} - \frac{\sqrt{2}}{8}\right)\right)$$
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)
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)