{"id":1849,"date":"2022-04-07T18:46:02","date_gmt":"2022-04-07T18:46:02","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/?post_type=chapter&p=1849"},"modified":"2022-04-07T18:46:05","modified_gmt":"2022-04-07T18:46:05","slug":"solving-radical-equations","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/chapter\/solving-radical-equations\/","title":{"raw":"Solving Radical Equations","rendered":"Solving Radical Equations"},"content":{"raw":"
\r\n

Learning Outcomes<\/h3>\r\n
\r\n
    \r\n \t
  • Solve a radical equation by isolating the radical term and then raising both sides to a power to remove the radical<\/li>\r\n \t
  • Identify a radical equation with no solution or extraneous solutions<\/li>\r\n<\/ul>\r\n<\/section><\/div>\r\nAn equation that contains a radical expression, such as a square root, is called a radical equation<\/strong>. Solving radical equations requires applying the rules of exponents and following some basic algebraic principles. In some cases, it also requires looking out for errors generated by raising unknown quantities to an even power.\r\n

    Isolate a Radical Term<\/h2>\r\nA basic strategy for solving radical equations is to isolate the radical term first, and then raise both sides of the equation to a power to remove the radical.\r\n\r\nThere are two key ideas that you will be using to solve radical equations. The first is\r\n

    if [latex] a=b[\/latex], then [latex] {{a}^{2}}={{b}^{2}}[\/latex].<\/p>\r\nThis property allows you to square both sides of an equation and remain certain that the two sides are still equal. The second is that if the square root of any nonnegative number x<\/i> is squared, then you get x.<\/i>\r\n

    For [latex]x \\geq 0, {{\\left( \\sqrt{x} \\right)}^{2}}=x[\/latex].<\/p>\r\nThis property allows you to \u201cremove\u201d the radicals from your equations.\r\n\r\nLet\u2019s start with a radical equation that can be solved in a few steps:\r\n

    \r\n

    Example<\/h3>\r\nSolve [latex]\\sqrt{x}=3[\/latex].\r\n\r\n[reveal-answer q=\"114881\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"114881\"]\r\n\r\nSquare each side of the equation to remove the radical.\r\n

    [latex](\\sqrt{x})^2 = 3^2[\/latex]<\/p>\r\nSimplify.\r\n

    [latex]x=9[\/latex]<\/p>\r\nCheck your solution: Since [latex]\\sqrt{9}=3[\/latex], the solution [latex]x=9[\/latex] is correct.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nLet\u2019s look at another example.\r\n

    \r\n

    Example<\/h3>\r\nSolve [latex]\\sqrt{2x}=4[\/latex].\r\n\r\n[reveal-answer q=\"542500\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"542500\"]\r\n\r\nSquare each side of the equation to remove the radical.\r\n

    [latex](\\sqrt{2x})^2 = 4^2[\/latex]<\/p>\r\nSimplify.\r\n

    [latex]2x=16[\/latex]<\/p>\r\nDivide each side of the equation by [latex]2[\/latex] to isolate [latex]x[\/latex].\r\n

    [latex]\\frac{2x}{2}=\\frac{16}{2}[\/latex]<\/p>\r\nSimplify.\r\n

    [latex]x=8[\/latex]<\/p>\r\nCheck your solution: Since [latex]\\sqrt{2 \\cdot 8} = \\sqrt{16} = 4[\/latex], the solution [latex]x=8[\/latex] is correct.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nIn these examples, the radical term appeared by itself on one side of the equation. If the equation contains additional terms on the side containing the radical, you must begin by isolating the radical. That is, get the radical by itself on one side of the equation.\r\n

    \r\n

    Example<\/h3>\r\nSolve [latex]2 \\sqrt{x} +3=13[\/latex].\r\n\r\n[reveal-answer q=\"114714\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"114714\"]\r\n\r\nIsolate the radical term by subtracting [latex]3[\/latex] from each side of the equation.\r\n

    [latex]2 \\sqrt{x}=10[\/latex]<\/p>\r\nIsolate the radical itself by dividing each side of the equation by [latex]2[\/latex].\r\n

    [latex]\\sqrt{x}=5[\/latex]<\/p>\r\nSquare each side of the equation to remove the radical.\r\n

    [latex](\\sqrt{x})^2 = 5^2[\/latex]<\/p>\r\nSimplify.\r\n

    [latex]x=25[\/latex]<\/p>\r\nCheck your solution: Since [latex]2 \\cdot \\sqrt{25} +3 = 2 \\cdot 5 + 3 = 10 + 3 =13[\/latex], the solution [latex]x=25[\/latex] is correct.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nIn the following video, you will see two more examples that are similar to the ones above.\r\n\r\nhttps:\/\/youtu.be\/tT0Zwsto6AQ\r\n

    \r\n

    Solving Radical Equations<\/h3>\r\nFollow the following four steps to solve radical equations.\r\n
      \r\n \t
    1. Isolate the radical expression.<\/li>\r\n \t
    2. Square both sides of the equation: If [latex]x=y[\/latex], then [latex]x^{2}=y^{2}[\/latex].<\/li>\r\n \t
    3. Once the radical is removed, solve for the unknown.<\/li>\r\n \t
    4. Check all answers.<\/li>\r\n<\/ol>\r\n<\/div>\r\n

      Identify a Radical Equation with No Solutions or Extraneous Solutions<\/h2>\r\nIt is important to check your solutions\u2014especially when solving radical equations. Look carefully at the next example.\r\n
      \r\n

      Example<\/h3>\r\nSolve [latex]\\sqrt{x}+5=2[\/latex].\r\n\r\n[reveal-answer q=\"233684\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"233684\"]\r\n\r\nIsolate the radical expression by subtracting [latex]5[\/latex] from each side.\r\n

      [latex]\\sqrt{x}=-3[\/latex]<\/p>\r\nSquare both side of the equation.\r\n

      [latex](\\sqrt{x})^2 = (-3)^2[\/latex]<\/p>\r\nOnce the radical is removed, solve for the unknown.\r\n

      [latex]x=9[\/latex]<\/p>\r\nCheck your solution.\r\n

      [latex]\\sqrt{9}+5=2[\/latex]<\/p>\r\n

      [latex]3+5=2[\/latex]<\/p>\r\n

      [latex]8 \\neq2[\/latex]<\/p>\r\nThis means that there is no solution to this equation.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nThe answer does not produce a true statement when substituted back into the original equation. Notice that the radical is isolated, we had [latex]\\sqrt{x}=-3[\/latex].\u00a0The principal square root of a number can only be nonnegative<\/em>. This means that no value for a will result in a radical expression whose positive square root is [latex]-3[\/latex].\r\n\r\nIncorrect values of the variable, such as those that are introduced as a result of the squaring process, are called extraneous solutions<\/strong>. Extraneous solutions can be identified because they will not create a true statement when substituted back into the original equation. This is one of the reasons why checking your work is so important\u2014if your answers does not satisfy the original equation you need to exclude it from the solution set.\r\n\r\nIn the following video, we present more examples of solving radical equations by isolating a radical term on one side.\r\n\r\nhttps:\/\/youtu.be\/qkZHKK77grM","rendered":"

      \n

      Learning Outcomes<\/h3>\n
      \n
        \n
      • Solve a radical equation by isolating the radical term and then raising both sides to a power to remove the radical<\/li>\n
      • Identify a radical equation with no solution or extraneous solutions<\/li>\n<\/ul>\n<\/section>\n<\/div>\n

        An equation that contains a radical expression, such as a square root, is called a radical equation<\/strong>. Solving radical equations requires applying the rules of exponents and following some basic algebraic principles. In some cases, it also requires looking out for errors generated by raising unknown quantities to an even power.<\/p>\n

        Isolate a Radical Term<\/h2>\n

        A basic strategy for solving radical equations is to isolate the radical term first, and then raise both sides of the equation to a power to remove the radical.<\/p>\n

        There are two key ideas that you will be using to solve radical equations. The first is<\/p>\n

        if [latex]a=b[\/latex], then [latex]{{a}^{2}}={{b}^{2}}[\/latex].<\/p>\n

        This property allows you to square both sides of an equation and remain certain that the two sides are still equal. The second is that if the square root of any nonnegative number x<\/i> is squared, then you get x.<\/i><\/p>\n

        For [latex]x \\geq 0, {{\\left( \\sqrt{x} \\right)}^{2}}=x[\/latex].<\/p>\n

        This property allows you to \u201cremove\u201d the radicals from your equations.<\/p>\n

        Let\u2019s start with a radical equation that can be solved in a few steps:<\/p>\n

        \n

        Example<\/h3>\n

        Solve [latex]\\sqrt{x}=3[\/latex].<\/p>\n

        Show Answer<\/span><\/p>\n
        \n

        Square each side of the equation to remove the radical.<\/p>\n

        [latex](\\sqrt{x})^2 = 3^2[\/latex]<\/p>\n

        Simplify.<\/p>\n

        [latex]x=9[\/latex]<\/p>\n

        Check your solution: Since [latex]\\sqrt{9}=3[\/latex], the solution [latex]x=9[\/latex] is correct.<\/p>\n<\/div>\n<\/div>\n<\/div>\n

        Let\u2019s look at another example.<\/p>\n

        \n

        Example<\/h3>\n

        Solve [latex]\\sqrt{2x}=4[\/latex].<\/p>\n

        Show Answer<\/span><\/p>\n
        \n

        Square each side of the equation to remove the radical.<\/p>\n

        [latex](\\sqrt{2x})^2 = 4^2[\/latex]<\/p>\n

        Simplify.<\/p>\n

        [latex]2x=16[\/latex]<\/p>\n

        Divide each side of the equation by [latex]2[\/latex] to isolate [latex]x[\/latex].<\/p>\n

        [latex]\\frac{2x}{2}=\\frac{16}{2}[\/latex]<\/p>\n

        Simplify.<\/p>\n

        [latex]x=8[\/latex]<\/p>\n

        Check your solution: Since [latex]\\sqrt{2 \\cdot 8} = \\sqrt{16} = 4[\/latex], the solution [latex]x=8[\/latex] is correct.<\/p>\n<\/div>\n<\/div>\n<\/div>\n

        In these examples, the radical term appeared by itself on one side of the equation. If the equation contains additional terms on the side containing the radical, you must begin by isolating the radical. That is, get the radical by itself on one side of the equation.<\/p>\n

        \n

        Example<\/h3>\n

        Solve [latex]2 \\sqrt{x} +3=13[\/latex].<\/p>\n

        Show Answer<\/span><\/p>\n
        \n

        Isolate the radical term by subtracting [latex]3[\/latex] from each side of the equation.<\/p>\n

        [latex]2 \\sqrt{x}=10[\/latex]<\/p>\n

        Isolate the radical itself by dividing each side of the equation by [latex]2[\/latex].<\/p>\n

        [latex]\\sqrt{x}=5[\/latex]<\/p>\n

        Square each side of the equation to remove the radical.<\/p>\n

        [latex](\\sqrt{x})^2 = 5^2[\/latex]<\/p>\n

        Simplify.<\/p>\n

        [latex]x=25[\/latex]<\/p>\n

        Check your solution: Since [latex]2 \\cdot \\sqrt{25} +3 = 2 \\cdot 5 + 3 = 10 + 3 =13[\/latex], the solution [latex]x=25[\/latex] is correct.<\/p>\n<\/div>\n<\/div>\n<\/div>\n

        In the following video, you will see two more examples that are similar to the ones above.<\/p>\n