Quadratic Factoring Calculator (AC Method)

📚 Why do we need two numbers that multiply to \( a \times c \) and add to \( b \)?

The AC Method (also called factoring by grouping) is based on a simple algebraic manipulation:

We want to factor \( ax^2 + bx + c \) into the form \( (px + q)(rx + s) \).

If we expand \( (px + q)(rx + s) \), we get:

\( (px + q)(rx + s) = pr\,x^2 + (ps + qr)x + qs \)

By comparing with \( ax^2 + bx + c \), we see:

• \( a = pr \) (product of the x-coefficients)
• \( c = qs \) (product of the constants)
• \( b = ps + qr \) (sum of the cross terms)

Now, notice that \( a \times c = (pr)(qs) = (ps)(qr) \).

If we let \( m = ps \) and \( n = qr \), then:

• \( m \times n = (ps)(qr) = a \times c \)
• \( m + n = ps + qr = b \)

Therefore, finding two numbers \( m \) and \( n \) that multiply to \( a \times c \) and add to \( b \) allows us to split the middle term \( bx \) into \( mx + nx \), then factor by grouping.

This method works when the quadratic is factorable over the rationals (i.e., when the discriminant \( \Delta = b^2 - 4ac \) is a perfect square).