Method of Transform

William F. Barnes
wfbarnes.xyz

Abstract

Factoring quadratics often relies on a numerical search for two numbers whose sum and product match given coefficients. In this note, that search is written algebraically through a related monic equation whose solutions determine the middle-term split. Solving this equation gives a direct route to the binomial factors while also connecting the logic behind standard methods such as the AC method and slide-and-divide.

DOI

10.1080/00255572.2026.2708216

This is the article’s permanent link and will become available when the article is published online.

Examples

Worked examples

A numbered companion PDF of examples using the method of transform.

Quick course

1. Begin with a quadratic expression in its usual form.

A x squared plus B x plus C

2. Splitting the middle term requires two numbers whose sum is B and whose product is AC.

w one plus w two equals B, and w one times w two equals A C

3. Instead of searching for those numbers blindly, treat them as the two solutions of a new quadratic in w.

w squared minus B w plus A C equals zero

4. This transformed equation is monic and is usually easier to solve. Its solutions, w1 and w2, give the middle-term split directly.

A x squared plus w one x plus w two x plus C

5. The four-term expression can then be grouped into the desired pair of linear factors.

A x squared plus B x plus C equals alpha x plus beta times alpha prime x plus beta prime

6. To reduce inspection, choose either solution w and use its greatest common divisor with A.

alpha equals the greatest common divisor of A and w

Then use β = αC/w, α′ = A/α, and β′ = C/β.