# Reducing Binary Quadratic Forms

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

**Reduce the binary quadratic form $f(x, y) = 4x^2 - 24xy +39y^2$.**

We have that $a = 4$, $b = -24$, and $c = 39$. So:

(1)There exists an integer $m$ such that:

(2)That is:

(3)Observe that $m = 3$ works. So an equivalent binary quadratic form is:

(4)Now we have:

(5)So applying $M_1$ to $f(x, y)$ gives us:

(6)So we have that $-3 < 0 < 3 < 4$, and so $f(x, y) = 3x^2 + 4y^2$ is a reduced binary quadratic form that is equivalent to $f(x, y) = 4x^2 - 24xy +39y^2$.

## Example 2

**Reduce the binary quadratic form $f(x, y) = 6x^2 - 5xy + 4y^2$.**

We have that:

(7)Applying $M_1$ to $f(x, y)$ gives us:

(8)Now we have that:

(9)There exists an integer $m$ such that:

(10)So $m = -1$. Applying $M_2$ with $m = -1$ and we get:

(11)We have that $-4 < -3 \leq -4 < 6$, and so $f(x, y) = 4x^2 - 3xy + 6y^2$ is a reduced binary quadratic form that is equivalent to $6x^2 - 5xy + 4y^2$.