Remove Two Radicals from a Linear Algebraic Equation

By Mary Jane Sterling

Some algebraic equations that contain radicals call for more than one application of squaring both sides. For example, you usually need to square both sides more than once when you have three terms in the equation — two of them with radicals.

For example, say you have to work with the equation

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Here’s how you solve the problem:

  1. Move the radicals so that only one appears on each side.

    image1.png

  2. Square both sides of the equation.

    image2.png

    After simplifying the results of the first two steps, you have the following:

    image3.png

  3. Move all the nonradical terms to the left and simplify.

    This gives you

    image4.png

  4. Make the job of squaring the binomial on the left easier by dividing each term by two — the common factor of all the terms on both sides.

    You end up with

    image5.png

    which becomes

    image6.png

  5. Square both sides again, simplify, set the quadratic equal to zero, and solve for x.

    This process gives you the following:

    image7.png

    When x – 2 = 0, x = 2; and when x – 34 = 0, x = 34.

  6. Don’t forget to check each solution in the original equation:

    image8.png

    The solution x = 2 works. The other solution, x = 34, doesn’t work in the equation. The number 34 is an extraneous solution.