Sure! Let's go step by step to understand and simplify the given equation:
We start with the equation:
[tex]\[ 15 x^2 + 4 x y - 4 y^2 - 10 x + 4 y = 0 \][/tex]
### Step 1: Identify the type of equation
This is a quadratic equation in two variables [tex]\(x\)[/tex] and [tex]\(y\)[/tex]. It contains terms involving [tex]\(x^2\)[/tex], [tex]\(y^2\)[/tex], [tex]\(xy\)[/tex], and linear terms in [tex]\(x\)[/tex] and [tex]\(y\)[/tex].
### Step 2: Rewrite the equation in standard quadratic form for conics
Rewrite the equation and group similar terms:
[tex]\[ 15 x^2 + 4 x y - 4 y^2 - 10 x + 4 y = 0 \][/tex]
### Step 3: Check if it can be further simplified or factored
Next, gather the information to understand its geometrical representation (such as whether it's a parabola, ellipse, hyperbola). Since our terms contain both [tex]\(x^2\)[/tex], [tex]\(y^2\)[/tex], as well as [tex]\(xy\)[/tex], it is likely to represent a hyperbola unless further simplification reveals otherwise.
### Step 4: Verify the simplified form (completing the square or diagonalization can be considered but were resolved to be complex)
No special simplification or elliptic simplification techniques have yielded a straightforward result beyond recognizing the quadratic form's complexity.
Therefore, we conclude that this is indeed a quadratic equation in the stated form, containing all significant terms indicative of its square representation's extended or possible conic nature.
Thus, the answer remains as already simplified:
[tex]\[15x^2 + 4xy - 10x - 4y^2 + 4y = 0\][/tex]
Since it's simplified, your next steps might involve specific analysis paths (like confirming a hyperbolic format) for predictive plotting or graphical representation. For now, the equation above stands as simplified and final.
If any specific nature exploration (hyperbolic, ellipse orientation) were needed, you'd look towards completing the square steps properly or pedagogic transformations.
