Factoring X Squared Minus 2

Depending on its order and the number of possessed terms, polynomial factorization can be a lengthy and complicated process. The polynomial expression, (​x​2 − 2), is fortunately not one of those polynomials. The expression (​x​2 − 2) is a classic example of a difference of two squares. In factoring a difference of two squares, any expression in the form of (​a​2 − ​b​2) is reduced to (​a​ − ​b​)(​a​ + ​b​). The key to this factoring process and ultimate solution for the expression (​x​2 − 2) lies in the square roots of its terms.

1. Calculating Square Roots

Calculate the square roots for 2 and ​x​2. The square root of 2 is √2 and the square root of ​x​2 is ​x​.

2. Factoring the Polynomial

Write the equation

((x^2-2))

as the difference of two squares employing the terms’ square roots. You find that

((x^2-2) = (x-sqrt{2}) (x+sqrt{2}))

3. Solving the Equation

Set each expression in parentheses equal to 0, then solve. The first expression set to 0 yields

((x-sqrt{2})=0 text{ therefore } x= sqrt{2})

The second expression set to 0 yields

((x+ sqrt{2}) = 0 text{ therefore } x=- sqrt{2})

The solutions for ​x​ are √2 and −√2.

TL;DR (Too Long; Didn’t Read)

If needed, √2 can be converted into decimal form with a calculator, resulting in 1.41421356.