Find three smallest non-zero values represented by the form 7x2−5xy+11y2
.


(5,7,11)
 

(5,7,13)
 

(5,11,13)
 

(7,9,11)
 

(7,9,13)
 

(7,11,13)

To solve the problem of finding the three smallest non-zero values represented by the quadratic form:

\[
7x^2 - 5xy + 11y^2
\]

We substitute integer values for \(x\) and \(y\) and compute the expression for each combination of small integers.

### Calculating for different combinations of \((x, y)\):

#### 1. \(x = 1, y = 0\):
\[
7(1)^2 - 5(1)(0) + 11(0)^2 = 7
\]

#### 2. \(x = 0, y = 1\):
\[
7(0)^2 - 5(0)(1) + 11(1)^2 = 11
\]

#### 3. \(x = 1, y = 1\):
\[
7(1)^2 - 5(1)(1) + 11(1)^2 = 7 - 5 + 11 = 13
\]

#### 4. \(x = 2, y = 1\):
\[
7(2)^2 - 5(2)(1) + 11(1)^2 = 7(4) - 5(2)(1) + 11 = 28 - 10 + 11 = 29
\]

#### 5. \(x = 1, y = 2\):
\[
7(1)^2 - 5(1)(2) + 11(2)^2 = 7 - 10 + 44 = 41
\]

#### 6. \(x = -1, y = 1\):
\[
7(-1)^2 - 5(-1)(1) + 11(1)^2 = 7 + 5 + 11 = 23
\]

### Smallest non-zero values:

From the calculations above, the three smallest non-zero values of the form \(7x^2 - 5xy + 11y^2\) are:

\[
7, 11, 13
\]

Therefore, the correct set of values is **\(7, 11, 13\)**, and this corresponds to the option **(7, 11, 13)**.

Would you like more details or have any questions?

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Here are 5 related questions:

1. How are quadratic forms classified in mathematics?
2. Can this method be applied to more complex quadratic forms?
3. What are the geometric interpretations of quadratic forms?
4. How do quadratic forms relate to matrices?
5. What are the applications of quadratic forms in physics and engineering?

**Tip:** When analyzing quadratic forms, checking for symmetries or simplifying assumptions can make the computations faster.

Find three smallest non-zero values represented by the form 7x^2 − 5xy + 11y^2.

Discover how to find the three smallest non-zero values represented by the quadratic form 7x^2 − 5xy + 11y^2. Detailed calculations and examples are provided, making this problem suitable for high school students (Grades 10-12).

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