<!DOCTYPE html>
<html lang="en">
<head>
  <meta charset="UTF-8">
  <meta name="viewport" content="width=device-width, initial-scale=1.0">
  <title>Why Extraneous Solutions Appear</title>
  <style>
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  </style>
</head>
<body>

  <div class="container">
    <div class="header-title">Why Extraneous Solutions Appear</div>

    <div class="panels">
      
      <div class="panel trap">
        <div class="panel-header">
          <span class="badge-num">1</span>
          <span>The Trap: Squaring</span>
        </div>

        <div class="funnel-wrap">
          <div class="funnel-inputs">
            <div class="funnel-pill math-font">
              (<span class="sign-neg">-4</span>)<sup>2</sup> = +16
            </div>
            <div class="funnel-pill math-font">
              (<span class="sign-pos">+4</span>)<sup>2</sup> = +16
            </div>
          </div>
          <div class="funnel-arrow">➜</div>
          <div class="funnel-output">
            <div class="step-label" style="margin-bottom: 2px;">Unified</div>
            <div class="output-val math-font">+16</div>
          </div>
        </div>

        <div class="panel-note">
          Squaring destroys sign information: both <strong>-4</strong> and <strong>+4</strong> yield <strong>+16</strong>.
        </div>
      </div>

      
      <div class="panel test">
        <div class="panel-header">
          <span class="badge-num">2</span>
          <span>The Test: Check x = 19</span>
        </div>

        <div class="steps-list">
          <div class="step-item">
            <span class="step-label">Original</span>
            <span class="step-math math-font">√(x - 3) = -4</span>
          </div>
          <div class="step-item">
            <span class="step-label">Substitute</span>
            <span class="step-math math-font">√(19 - 3) = √16 = +4</span>
          </div>
        </div>

        <div class="alert-mismatch">
          <span style="font-size: 16px;">⚠️</span>
          <span class="mismatch-text math-font">+4 ≠ -4 (False!)</span>
        </div>

        <div class="tag-extraneous">
          Extraneous / No Real Solution
        </div>
      </div>
    </div>
  </div>

</body>
</html>

Solving Square Root Equations with Isolated Radicals

<!DOCTYPE html>
<html lang="en">
<head>
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</head>
<body>

  <div class="nav-bar">
    <button class="btn" id="prevBtn" onclick="prevStep()">◀ Prev</button>
    <div class="dots">
      <div class="dot active" onclick="goToStep(0)" title="Step 1: Solve Equation"></div>
      <div class="dot" onclick="goToStep(1)" title="Step 2: Check x = 6"></div>
      <div class="dot" onclick="goToStep(2)" title="Step 3: Check x = -3"></div>
    </div>
    <span class="step-indicator" id="stepIndicator">1 / 3</span>
    <button class="btn" id="nextBtn" onclick="nextStep()">Next ▶</button>
  </div>

  <div class="diagram-flow">
    
    <div class="top-stack" onclick="goToStep(0)" style="cursor: pointer;">
      <div class="math-card" id="step1Card">
        <div class="card-label">Given Radical Equation &amp; Square Both Sides</div>
        <div class="math-text" id="eqRoot">&radic;(3x + 18) = x</div>
        <div class="math-subtext" id="eqSquare">(&radic;(3x + 18))&sup2; = x&sup2; &rArr; x&sup2; &minus; 3x &minus; 18 = 0</div>
      </div>

      <div class="math-card" id="step2Card">
        <div class="card-label">Factor to Find Potential Solutions</div>
        <div class="math-text" id="eqFactor">(x &minus; 6)(x + 3) = 0</div>
      </div>
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    <svg class="arrow-svg" viewBox="0 0 200 20" preserveAspectRatio="none">
      <path d="M 100 0 L 100 6 L 50 6 L 50 16" fill="none" stroke="#64748b" stroke-width="1.5" />
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      <div class="branch-card" id="branchLeft" onclick="goToStep(1)">
        <div class="branch-root" id="rootLeft">x = 6</div>
        <div class="test-box">
          <div class="test-box-label">Substitute into original:</div>
          <div id="testLeft">&radic;(3(6) + 18) = &radic;36 = 6</div>
        </div>
        <div class="check-row valid">
          <span id="evalLeft">6 = 6</span>
          <span>&#10003; True</span>
        </div>
        <span class="badge badge-green">Valid Solution</span>
      </div>

      
      <div class="branch-card" id="branchRight" onclick="goToStep(2)">
        <div class="branch-root" id="rootRight">x = &minus;3</div>
        <div class="test-box">
          <div class="test-box-label">Substitute into original:</div>
          <div id="testRight">&radic;(3(&minus;3) + 18) = &radic;9 = 3</div>
        </div>
        <div class="check-row extraneous">
          <span id="evalRight">3 &ne; &minus;3</span>
          <span>&#10007; False</span>
        </div>
        <span class="badge badge-red">Extraneous</span>
      </div>
    </div>
  </div>

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          katex.render("(\\sqrt{3x + 18})^2 = x^2 \\implies x^2 - 3x - 18 = 0", document.getElementById('eqSquare'));
          katex.render("(x - 6)(x + 3) = 0", document.getElementById('eqFactor'));

          katex.render("x = 6", document.getElementById('rootLeft'));
          katex.render("\\sqrt{3(6) + 18} = \\sqrt{36} = 6", document.getElementById('testLeft'));
          katex.render("6 = 6", document.getElementById('evalLeft'));

          katex.render("x = -3", document.getElementById('rootRight'));
          katex.render("\\sqrt{3(-3) + 18} = \\sqrt{9} = 3", document.getElementById('testRight'));
          katex.render("3 \\neq -3", document.getElementById('evalRight'));
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</body>
</html>

<!DOCTYPE html>
<html lang="en">
<head>
  <meta charset="UTF-8">
  <meta name="viewport" content="width=device-width, initial-scale=1.0">
  <meta name="aspect-ratio" content="4:3">
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<body>
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Solving Square Root Equations with Non-Isolated Radicals

Solving Square Root Equations with Variables on Both Sides

OGM.10.REI.3

Aligned learning resources

Related diagrams