How to Factor X Squared Minus 2

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Depending on its order and the number of possessed terms, polynomial factorization can be a lengthy and complicated process. The polynomial expression, (​x2 − 2), is fortunately not one of those polynomials. The expression (​x2 − 2) is a classic example of a difference of two squares. In factoring a difference of two squares, any expression in the form of (​a2 − ​b2) is reduced to (​a​ − ​b​)(​a​ + ​b​). The key to this factoring process and ultimate solution for the expression (​x2 − 2) lies in the square roots of its terms.

  1. Calculating Square Roots

  2. Calculate the square roots for 2 and ​x2. The square root of 2 is √2 and the square root of ​x2 is ​x​.

  3. Factoring the Polynomial

  4. Write the equation


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

    (x^2-2) = (x-\sqrt{2}) (x+\sqrt{2})
  5. Solving the Equation

  6. 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.


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



About the Author

Chance E. Gartneer began writing professionally in 2008 working in conjunction with FEMA. He has the unofficial record for the most undergraduate hours at the University of Texas at Austin. When not working on his children's book masterpiece, he writes educational pieces focusing on early mathematics and ESL topics.