Find three smallest non-zero values represented by the form 5x^2−3xy+7y^2

We are asked to find the three smallest non-zero values represented by the quadratic form \( Q(x, y) = 5x^2 - 3xy + 7y^2 \). The approach involves testing small integer values for \(x\) and \(y\) and calculating \(Q(x, y)\) to determine the smallest non-zero values.

### Steps to approach:
1. **Define the quadratic form**:  
   \[
   Q(x, y) = 5x^2 - 3xy + 7y^2
   \]
2. **Test small integer values** of \(x\) and \(y\). We will start with combinations of \(x, y \in \{-2, -1, 0, 1, 2\}\).

Let's calculate \(Q(x, y)\) for various pairs of \(x\) and \(y\).

### Calculation of \(Q(x, y)\):

#### When \( x = 1 \):
- \( Q(1, 0) = 5(1)^2 + 0 = 5 \)
- \( Q(1, 1) = 5(1)^2 - 3(1)(1) + 7(1)^2 = 5 - 3 + 7 = 9 \)
- \( Q(1, -1) = 5(1)^2 - 3(1)(-1) + 7(-1)^2 = 5 + 3 + 7 = 15 \)

#### When \( x = -1 \):
- \( Q(-1, 0) = 5(-1)^2 + 0 = 5 \)
- \( Q(-1, 1) = 5(-1)^2 - 3(-1)(1) + 7(1)^2 = 5 + 3 + 7 = 15 \)
- \( Q(-1, -1) = 5(-1)^2 - 3(-1)(-1) + 7(-1)^2 = 5 - 3 + 7 = 9 \)

#### When \( x = 2 \):
- \( Q(2, 0) = 5(2)^2 + 0 = 20 \)
- \( Q(2, 1) = 5(2)^2 - 3(2)(1) + 7(1)^2 = 20 - 6 + 7 = 21 \)
- \( Q(2, -1) = 5(2)^2 - 3(2)(-1) + 7(-1)^2 = 20 + 6 + 7 = 33 \)

#### When \( x = -2 \):
- \( Q(-2, 0) = 5(-2)^2 + 0 = 20 \)
- \( Q(-2, 1) = 5(-2)^2 - 3(-2)(1) + 7(1)^2 = 20 + 6 + 7 = 33 \)
- \( Q(-2, -1) = 5(-2)^2 - 3(-2)(-1) + 7(-1)^2 = 20 - 6 + 7 = 21 \)

### Identifying the smallest non-zero values:
From the calculations, the distinct non-zero values of \(Q(x, y)\) are:
- \(5\), \(9\), \(15\), \(20\), \(21\), \(33\)

Thus, the **three smallest non-zero values** are:
\[
\boxed{5, 9, 15}
\]

Would you like to explore more details or ask any questions?

### Related Questions:
1. How does the quadratic form \(5x^2 - 3xy + 7y^2\) behave for larger values of \(x\) and \(y\)?
2. Can we find the minimum possible value of \(Q(x, y)\)?
3. What are the properties of quadratic forms in general?
4. How does the discriminant of the quadratic form affect its properties?
5. Can this form be transformed into a simpler canonical form?

### Tip:
In problems involving quadratic forms, symmetry or specific patterns in the coefficients can help reduce computation efforts.

Learn how to find the three smallest non-zero values represented by the quadratic form 5x^2 - 3xy + 7y^2. Detailed step-by-step calculation for small integer values of x and y. Suitable for students in Grades 10-12.

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