The equation given in the image is:

$5x^2 - 6xy + 5y^2 - 6\sqrt{2}x + 2\sqrt{2}y - 4 = 0$

The task is to simplify this equation. To solve this, let's try completing the square for the terms involving $x$ and $y$.

Step-by-Step Simplification:

1. Group the quadratic and linear terms involving $x$ and $y$: $5x^2 - 6xy + 5y^2 - 6\sqrt{2}x + 2\sqrt{2}y - 4$

2. Notice that the quadratic terms $5x^2 - 6xy + 5y^2$ can be written as: $(x - y)^2 + 4(x - y)y$

Continue simplificationLet's systematically simplify the equation. We aim to rewrite it in a more manageable form by completing the square for the terms involving $x$ and $y$.

The equation is:

$5x^2 - 6xy + 5y^2 - 6\sqrt{2}x + 2\sqrt{2}y - 4 = 0$

Step 1: Factorize the quadratic terms

The terms $5x^2 - 6xy + 5y^2$ can be rewritten using a quadratic expression in two variables: $5(x^2 - \frac{6}{5}xy + y^2).$

Inside the parentheses, we complete the square for $x^2 - \frac{6}{5}xy + y^2$. The coefficient of $xy$ is $-\frac{6}{5}$, and half of it squared is: $\left(\frac{-\frac{6}{5}}{2}\right)^2 = \left(-\frac{3}{5}\right)^2 = \frac{9}{25}.$

Thus, we rewrite: $x^2 - \frac{6}{5}xy + y^2 = \left(x - \frac{3}{5}y\right)^2 - \frac{9}{25}y^2.$

So, the quadratic terms become: $5\left[\left(x - \frac{3}{5}y\right)^2 - \frac{9}{25}y^2\right] = 5\left(x - \frac{3}{5}y\right)^2 - \frac{9}{5}y^2.$

The equation is now: $5\left(x - \frac{3}{5}y\right)^2 - \frac{9}{5}y^2 - 6\sqrt{2}x + 2\sqrt{2}y - 4 = 0.$

Step 2: Handle the linear terms

Group the linear terms $-6\sqrt{2}x + 2\sqrt{2}y$ and rewrite them in terms of $x - \frac{3}{5}y$.

Substitute $z = x - \frac{3}{5}y$. This makes $x = z + \frac{3}{5}y$. Substituting back into $-6\sqrt{2}x + 2\sqrt{2}y$, we get: $-6\sqrt{2}x + 2\sqrt{2}y = -6\sqrt{2}(z + \frac{3}{5}y) + 2\sqrt{2}y.$

Expanding: $-6\sqrt{2}z - \frac{18\sqrt{2}}{5}y + 2\sqrt{2}y = -6\sqrt{2}z - \frac{8\sqrt{2}}{5}y.$

Step 3: Rewrite the equation

Substitute everything back into the equation. It becomes: $5z^2 - \frac{9}{5}y^2 - 6\sqrt{2}z - \frac{8\sqrt{2}}{5}y - 4 = 0.$

This is now simplified and can be solved further if necessary.

Do you want to proceed with finding specific solutions or further rewriting? Let me know!

Related Questions:

1. How do we solve this quadratic equation in terms of $x$ and $y$?
2. What is the geometric interpretation of this equation?
3. Can we represent this equation in matrix form?
4. How does completing the square help in solving quadratic equations?
5. How do mixed terms like $-6xy$ affect the simplification?

Tip:

Always start by grouping similar terms (e.g., quadratic, linear, constants) before attempting simplification.