Po-Shen Loh’s Method of Solving Quadratic Equation

Po-Shen Loh

Quadratic Equations:

Quadratic equations are those algebraic equations which are in the standard form of

where a, b and c are real and known coefficients with 𝑎 ≠ 0 and x is an unknown number. A quadratic equation is also called a second-degree equation since its highest exponent is 2. Solving a quadratic equation means finding the values for x that satisfy the equation, which are known as roots.

A quadratic equation can be expressed as

where 𝑝 and 𝑞 are the roots of the equation.

There are several ways to solve a quadratic equation, like factoring, graphing, completing square and so on. There is also a quadratic formula, 𝑥 = (−𝑏 ±√𝑏^2−4𝑎𝑐)/2𝑎 which directly gives the solutions.

In this article, I shall be writing about a different approach to solving quadratic equations. It’s called “Po-Shen Loh’s method”, or also “Babylonian’s method”. Po-Shen Loh is a professor of mathematics at Carnegie Mellon University and the national coach of the United States’ International Math Olympiad team. He came up with this method in 2019.

Derivation:

Firstly, to use Po-Shen Loh’s method, the leading coefficient should be 1 i.e. 𝑎 = 1 in equation (1). So, equation (2) becomes

where 𝐵 = 𝑏/𝑎 and 𝐶 = 𝑐/𝑎 .

From (𝑥 − 𝑝)(𝑥 − 𝑞), we get 𝑥^2 − (𝑝 + 𝑞)𝑥 + 𝑝𝑞.

Average of roots = (𝑝+𝑞)/2 = −𝐵/2

⇒ −𝐵 / 2 is the mid-number between 𝑝 and 𝑞. In other words, −𝐵 / 2 is equally distanced from 𝑝 and 𝑞. Let that distance be 𝑚.

⇒ The values of 𝑝 and 𝑞 are in the form −𝐵/2 ± 𝑚

From (4) and (5),

Lastly, substituting 𝑚 from (6) to (5), we get

And this is what we’re solving for.

Quadratic formula:

In equation (7), replacing 𝐵 and 𝐶 in terms of 𝑎, 𝑏 and 𝑐 from 𝑎𝑥^2 + 𝑏𝑥 + c

Reference:

Loh, P.S. 2019. A Simple Proof of the Quadratic Formula