𝟓𝒙𝒚𝒑𝟐+(𝒙𝟐−𝟓𝒚𝟐−𝟕)𝒑−𝒙𝒚=𝟎 Obtain complete primitive and singular solutions, also comments on geometry of singular solutions.

To solve the equation \( 5xyp^2 + (x^2 - 5y^2 - 7)p - xy = 0 \) for the complete primitive and singular solutions, we can proceed as follows:

### Step 1: Understand the General Form
The given equation is a quadratic equation in \( p \):
\[ 5xyp^2 + (x^2 - 5y^2 - 7)p - xy = 0 \]

### Step 2: Solve the Quadratic Equation for \( p \)
Using the quadratic formula \( p = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), where \( a = 5xy \), \( b = x^2 - 5y^2 - 7 \), and \( c = -xy \):

\[ p = \frac{-(x^2 - 5y^2 - 7) \pm \sqrt{(x^2 - 5y^2 - 7)^2 - 4 \cdot 5xy \cdot (-xy)}}{2 \cdot 5xy} \]

### Step 3: Simplify the Discriminant
The discriminant \( \Delta \) is:
\[ \Delta = (x^2 - 5y^2 - 7)^2 + 20x^2y^2 \]

### Step 4: Evaluate and Simplify the Solutions
Now plug the discriminant back into the quadratic formula:
\[ p = \frac{-(x^2 - 5y^2 - 7) \pm \sqrt{(x^2 - 5y^2 - 7)^2 + 20x^2y^2}}{10xy} \]

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Learn how to solve the quadratic equation 5xyp^2 + (x^2 - 5y^2 - 7)p - xy = 0 with a detailed step-by-step solution. This problem involves quadratic equations, the discriminant, and the quadratic formula. Suitable for high school students studying algebra and quadratic equations.

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