# Examples of Reducing Binary Quadratic Forms 2

Recall from the Reduced Binary Quadratic Forms page that a positive definite binary quadratic form $f(x, y) = ax^2 + bxy + cy^2$ is said to be reduced if either:

**1)**$-a < b \leq a < c$.

**2)**$0 < b < a = c$.

Also recall that $M_1 = \begin{bmatrix} 0 & -1\\ 1 & 0 \end{bmatrix}$ and $M_2 = M_2 = \begin{bmatrix} 1 & m\\ 0 & 1 \end{bmatrix}$ and if $a > c$ then applying $M_1$ to $f(x, y)$ gives us an equivalent binary quadratic form with $a < c$. If $b > a$ (equivalently $-a > b$) then applying $M_2$ to $f(x, y)$ gives us an equivalent binary quadratic form with $b < a$ (equivalently $-a < b$). We will now look at some examples of taking a binary quadratic form and finding an equivalent binary quadratic form that is reduced.

## Example 1

**Reduce the binary quadratic form $g(x, y) = 5x^2 - 6xy +6y^2$.**

We have that $a = 5$, $b = -6$, and $c = 6$. So:

(1)So we want to find the integer $m$ such that:

(2)So $m = 1$. Therefore using the matrix $M_2 = \begin{bmatrix} 1 & m \\ 0 & 1 \end{bmatrix} = \begin{bmatrix} 1 & 1 \\ 0 & 1 \end{bmatrix}$ to get:

(3)Observe that $f(x, y)$ is such that $0 < 4 < 5 = 5$, and so $f(x, y)$ is equivalent to the reduced binary quadratic form $g(x, y) = 5x^2 + 4xy + 5y^2$.

## Example 2

**Reduce the binary quadratic form $h(x, y) = 11x^2 - 2xy + 2y^2$.**

We have that $a = 11$, $b = -2$, and $c = 2$. So $-11 < -2 \leq 11 \not < 2$.

We apply the matrix $M_1 = \begin{bmatrix} 0 & -1\\ 1 & 0 \end{bmatrix}$ to get:

(4)Observe that $g(x, y)$ is such that $-2 < 2 \leq 2 < 11$. Therefore $g(x, y)$ is a reduced binary quadratic form that is equivalent to $h(x, y)$.