How To Factor 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.