4. Gira la cónica $$ 6 x^{2}-14 x y-8 y^{2}-16=0 $$ de manera que ya no tenga término en $$ x y $$. Escribe los coeficientes redondeando a dos decimales.

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Simplify the expression by following steps:
- step0: Evaluate the power:
  $$\left(\frac{\sqrt{2+\sqrt{2}}}{2}\right)^{2}$$
- step1: Evaluate the power:
  $$\frac{2+\sqrt{2}}{4}$$
Expand the expression $$ (\sqrt(2 - \sqrt(2))/2)^2 $$
Simplify the expression by following steps:
- step0: Evaluate the power:
  $$\left(\frac{\sqrt{2-\sqrt{2}}}{2}\right)^{2}$$
- step1: Evaluate the power:
  $$\frac{2-\sqrt{2}}{4}$$
Expand the expression $$ 6 * ((\sqrt(2 + \sqrt(2))/2) * x - (\sqrt(2 - \sqrt(2))/2) * y)^2 - 14 * ((\sqrt(2 + \sqrt(2))/2) * x - (\sqrt(2 - \sqrt(2))/2) * y) * ((\sqrt(2 - \sqrt(2))/2) * x + (\sqrt(2 + \sqrt(2))/2) * y) - 8 * ((\sqrt(2 - \sqrt(2))/2) * x + (\sqrt(2 + \sqrt(2))/2) * y)^2 - 16 $$
Simplify the expression by following steps:
- step0: Simplify:
  $$6\left(\left(\frac{\sqrt{2+\sqrt{2}}}{2}\right)x-\left(\frac{\sqrt{2-\sqrt{2}}}{2}\right)y\right)^{2}-14\left(\left(\frac{\sqrt{2+\sqrt{2}}}{2}\right)x-\left(\frac{\sqrt{2-\sqrt{2}}}{2}\right)y\right)\left(\left(\frac{\sqrt{2-\sqrt{2}}}{2}\right)x+\left(\frac{\sqrt{2+\sqrt{2}}}{2}\right)y\right)-8\left(\left(\frac{\sqrt{2-\sqrt{2}}}{2}\right)x+\left(\frac{\sqrt{2+\sqrt{2}}}{2}\right)y\right)^{2}-16$$
- step1: Remove the parentheses:
  $$6\left(\frac{\sqrt{2+\sqrt{2}}}{2}x-\left(\frac{\sqrt{2-\sqrt{2}}}{2}\right)y\right)^{2}-14\left(\left(\frac{\sqrt{2+\sqrt{2}}}{2}\right)x-\left(\frac{\sqrt{2-\sqrt{2}}}{2}\right)y\right)\left(\left(\frac{\sqrt{2-\sqrt{2}}}{2}\right)x+\left(\frac{\sqrt{2+\sqrt{2}}}{2}\right)y\right)-8\left(\left(\frac{\sqrt{2-\sqrt{2}}}{2}\right)x+\left(\frac{\sqrt{2+\sqrt{2}}}{2}\right)y\right)^{2}-16$$
- step2: Remove the parentheses:
  $$6\left(\frac{\sqrt{2+\sqrt{2}}}{2}x-\frac{\sqrt{2-\sqrt{2}}}{2}y\right)^{2}-14\left(\left(\frac{\sqrt{2+\sqrt{2}}}{2}\right)x-\left(\frac{\sqrt{2-\sqrt{2}}}{2}\right)y\right)\left(\left(\frac{\sqrt{2-\sqrt{2}}}{2}\right)x+\left(\frac{\sqrt{2+\sqrt{2}}}{2}\right)y\right)-8\left(\left(\frac{\sqrt{2-\sqrt{2}}}{2}\right)x+\left(\frac{\sqrt{2+\sqrt{2}}}{2}\right)y\right)^{2}-16$$
- step3: Remove the parentheses:
  $$6\left(\frac{\sqrt{2+\sqrt{2}}}{2}x-\frac{\sqrt{2-\sqrt{2}}}{2}y\right)^{2}-14\left(\left(\frac{\sqrt{2+\sqrt{2}}}{2}\right)x-\left(\frac{\sqrt{2-\sqrt{2}}}{2}\right)y\right)\left(\left(\frac{\sqrt{2-\sqrt{2}}}{2}\right)x+\left(\frac{\sqrt{2+\sqrt{2}}}{2}\right)y\right)-8\left(\frac{\sqrt{2-\sqrt{2}}}{2}x+\left(\frac{\sqrt{2+\sqrt{2}}}{2}\right)y\right)^{2}-16$$
- step4: Remove the parentheses:
  $$6\left(\frac{\sqrt{2+\sqrt{2}}}{2}x-\frac{\sqrt{2-\sqrt{2}}}{2}y\right)^{2}-14\left(\left(\frac{\sqrt{2+\sqrt{2}}}{2}\right)x-\left(\frac{\sqrt{2-\sqrt{2}}}{2}\right)y\right)\left(\left(\frac{\sqrt{2-\sqrt{2}}}{2}\right)x+\left(\frac{\sqrt{2+\sqrt{2}}}{2}\right)y\right)-8\left(\frac{\sqrt{2-\sqrt{2}}}{2}x+\frac{\sqrt{2+\sqrt{2}}}{2}y\right)^{2}-16$$
- step5: Remove the parentheses:
  $$6\left(\frac{\sqrt{2+\sqrt{2}}}{2}x-\frac{\sqrt{2-\sqrt{2}}}{2}y\right)^{2}-14\left(\frac{\sqrt{2+\sqrt{2}}}{2}x-\left(\frac{\sqrt{2-\sqrt{2}}}{2}\right)y\right)\left(\left(\frac{\sqrt{2-\sqrt{2}}}{2}\right)x+\left(\frac{\sqrt{2+\sqrt{2}}}{2}\right)y\right)-8\left(\frac{\sqrt{2-\sqrt{2}}}{2}x+\frac{\sqrt{2+\sqrt{2}}}{2}y\right)^{2}-16$$
- step6: Remove the parentheses:
  $$6\left(\frac{\sqrt{2+\sqrt{2}}}{2}x-\frac{\sqrt{2-\sqrt{2}}}{2}y\right)^{2}-14\left(\frac{\sqrt{2+\sqrt{2}}}{2}x-\frac{\sqrt{2-\sqrt{2}}}{2}y\right)\left(\left(\frac{\sqrt{2-\sqrt{2}}}{2}\right)x+\left(\frac{\sqrt{2+\sqrt{2}}}{2}\right)y\right)-8\left(\frac{\sqrt{2-\sqrt{2}}}{2}x+\frac{\sqrt{2+\sqrt{2}}}{2}y\right)^{2}-16$$
- step7: Remove the parentheses:
  $$6\left(\frac{\sqrt{2+\sqrt{2}}}{2}x-\frac{\sqrt{2-\sqrt{2}}}{2}y\right)^{2}-14\left(\frac{\sqrt{2+\sqrt{2}}}{2}x-\frac{\sqrt{2-\sqrt{2}}}{2}y\right)\left(\frac{\sqrt{2-\sqrt{2}}}{2}x+\left(\frac{\sqrt{2+\sqrt{2}}}{2}\right)y\right)-8\left(\frac{\sqrt{2-\sqrt{2}}}{2}x+\frac{\sqrt{2+\sqrt{2}}}{2}y\right)^{2}-16$$
- step8: Remove the parentheses:
  $$6\left(\frac{\sqrt{2+\sqrt{2}}}{2}x-\frac{\sqrt{2-\sqrt{2}}}{2}y\right)^{2}-14\left(\frac{\sqrt{2+\sqrt{2}}}{2}x-\frac{\sqrt{2-\sqrt{2}}}{2}y\right)\left(\frac{\sqrt{2-\sqrt{2}}}{2}x+\frac{\sqrt{2+\sqrt{2}}}{2}y\right)-8\left(\frac{\sqrt{2-\sqrt{2}}}{2}x+\frac{\sqrt{2+\sqrt{2}}}{2}y\right)^{2}-16$$
- step9: Expand the expression:
  $$\frac{6+3\sqrt{2}}{2}\times x^{2}-3\sqrt{2}\times xy+\frac{6-3\sqrt{2}}{2}\times y^{2}-14\left(\frac{\sqrt{2+\sqrt{2}}}{2}x-\frac{\sqrt{2-\sqrt{2}}}{2}y\right)\left(\frac{\sqrt{2-\sqrt{2}}}{2}x+\frac{\sqrt{2+\sqrt{2}}}{2}y\right)-8\left(\frac{\sqrt{2-\sqrt{2}}}{2}x+\frac{\sqrt{2+\sqrt{2}}}{2}y\right)^{2}-16$$
- step10: Expand the expression:
  $$\frac{6+3\sqrt{2}}{2}\times x^{2}-3\sqrt{2}\times xy+\frac{6-3\sqrt{2}}{2}\times y^{2}-\frac{7\sqrt{2}}{2}x^{2}-7\sqrt{2}\times xy+\frac{7\sqrt{2}}{2}y^{2}-8\left(\frac{\sqrt{2-\sqrt{2}}}{2}x+\frac{\sqrt{2+\sqrt{2}}}{2}y\right)^{2}-16$$
- step11: Expand the expression:
  $$\frac{6+3\sqrt{2}}{2}\times x^{2}-3\sqrt{2}\times xy+\frac{6-3\sqrt{2}}{2}\times y^{2}-\frac{7\sqrt{2}}{2}x^{2}-7\sqrt{2}\times xy+\frac{7\sqrt{2}}{2}y^{2}+\left(-4+2\sqrt{2}\right)x^{2}-4\sqrt{2}\times xy+\left(-4-2\sqrt{2}\right)y^{2}-16$$
- step12: Subtract the terms:
  $$-x^{2}-14\sqrt{2}\times xy-y^{2}-16$$
Después de realizar la rotación y simplificación de la ecuación, obtenemos la nueva forma de la cónica:

$$
-x^{2} - 14\sqrt{2} \cdot xy - y^{2} - 16 = 0
$$

### Paso 6: Coeficientes finales

Los nuevos coeficientes son:

- $$A' = -1$$
- $$B' = 0$$ (ya que no hay término $$xy$$)
- $$C' = -1$$

### Redondeo de los coeficientes

Los coeficientes $$A'$$ y $$C'$$ son:

- $$A' \approx -1.00$$
- $$C' \approx -1.00$$

Por lo tanto, los coeficientes redondeados a dos decimales son:

- $$A' = -1.00$$
- $$C' = -1.00$$

La ecuación final de la cónica, sin el término $$xy$$, es:

$$
-1.00 x^{2} - 1.00 y^{2} - 16 = 0
$$
Para eliminar el término $$xy$$ de la cónica $$6x^{2} - 14xy - 8y^{2} - 16 = 0$$, se realiza una rotación de ejes. Después de la rotación, los nuevos coeficientes son $$A' = -1.00$$ y $$C' = -1.00$$. La ecuación simplificada es $$-1.00x^{2} - 1.00y^{2} - 16 = 0$$.

Gira la cónica

de manera que ya no tenga término en . Escribe
los coeficientes redondeando a dos decimales.

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Gira la cónica de manera que ya no tenga término en . Escribe los coeficientes redondeando a dos decimales.