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Tutoring abilities

If someone tells you, "I took the square root of my secret number and got $6$," you can instantly read their mind.

To undo a square root, you perform its opposite operation: **squaring**.

Square roots and squaring are **inverse operations**, which means they directly cancel each other out.

Because the square root asks "what number multiplied by itself gives $x$," squaring that result brings you right back to $x$: $(\sqrt{x})^2 = x$.

<!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">
  <link rel="stylesheet" href="{{VENDOR_CDN}}/katex.min.css">
  <script src="{{VENDOR_CDN}}/katex.min.js"></script>
  <script src="{{VENDOR_CDN}}/classace-diagram.js"></script>
  <style>
    :root {
      --primary: #8b5cf6;
      --primary-light: #ede9fe;
      --accent: #10b981;
      --accent-light: #d1fae5;
      --text: #1e2945;
      --muted: #64748b;
      --border: #e2e8f0;
      --card-bg: #ffffff;
    }

    * {
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      margin: 0;
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      font-family: system-ui, -apple-system, BlinkMacSystemFont, "Segoe UI", Roboto, sans-serif;
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    body {
      background: transparent;
      color: var(--text);
      display: flex;
      flex-direction: column;
      align-items: center;
      justify-content: center;
      padding: 12px;
      min-height: 100%;
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    .main-layout {
      width: 100%;
      max-width: 540px;
      display: flex;
      flex-direction: column;
      gap: 12px;
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    /* Equation selector pills */
    .selector-bar {
      display: flex;
      gap: 8px;
      background: #f1f5f9;
      padding: 4px;
      border-radius: 24px;
    }

    .preset-btn {
      border: none;
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      padding: 6px 14px;
      border-radius: 18px;
      font-size: 14px;
      font-weight: 600;
      color: var(--muted);
      cursor: pointer;
      transition: all 0.2s ease;
    }

    .preset-btn.active {
      background: var(--card-bg);
      color: var(--primary);
      box-shadow: 0 1px 3px rgba(0,0,0,0.1);
    }

    /* Workspace */
    .workspace {
      width: 100%;
      display: grid;
      grid-template-columns: 1.1fr 0.9fr;
      gap: 12px;
      align-items: stretch;
    }

    @media (max-width: 440px) {
      .workspace {
        grid-template-columns: 1fr;
      }
    }

    .panel {
      background: var(--card-bg);
      border: 1px solid var(--border);
      border-radius: 12px;
      padding: 14px;
      display: flex;
      flex-direction: column;
      justify-content: center;
      align-items: center;
      min-height: 190px;
      position: relative;
    }

    /* Algebraic steps panel */
    .algebra-flow {
      width: 100%;
      display: flex;
      flex-direction: column;
      gap: 8px;
      align-items: center;
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    .step-row {
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    .step-row.active {
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    .step-badge {
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    .step-row.resolved .step-badge {
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    .math-expr {
      font-size: 16px;
    }

    /* Geometric square area visual */
    .geo-container {
      width: 100%;
      height: 100%;
      display: flex;
      flex-direction: column;
      align-items: center;
      justify-content: center;
      gap: 8px;
    }

    .square-box {
      width: 110px;
      height: 110px;
      background: var(--primary-light);
      border: 2px solid var(--primary);
      border-radius: 8px;
      display: flex;
      flex-direction: column;
      align-items: center;
      justify-content: center;
      position: relative;
      transition: background 0.3s ease, border-color 0.3s ease;
    }

    .square-box.solved {
      background: var(--accent-light);
      border-color: var(--accent);
    }

    .side-label-top {
      position: absolute;
      top: -22px;
      left: 50%;
      transform: translateX(-50%);
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      font-weight: 600;
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    .side-label-left {
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      top: 50%;
      transform: translateY(-50%) rotate(-90deg);
      font-size: 13px;
      font-weight: 600;
      color: var(--primary);
      white-space: nowrap;
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    .area-label {
      font-size: 15px;
      font-weight: 700;
      color: var(--text);
      text-align: center;
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    /* Step Controller */
    .controls {
      display: flex;
      align-items: center;
      gap: 8px;
    }

    .nav-btn {
      background: var(--card-bg);
      border: 1px solid var(--border);
      color: var(--text);
      width: 32px;
      height: 32px;
      border-radius: 50%;
      display: flex;
      align-items: center;
      justify-content: center;
      cursor: pointer;
      font-size: 13px;
      transition: all 0.2s ease;
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    .nav-btn:hover {
      border-color: var(--primary);
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      gap: 6px;
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    .dot {
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      background: var(--border);
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      transition: all 0.2s ease;
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    .dot.active {
      background: var(--primary);
      transform: scale(1.2);
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  </style>
</head>
<body>

  <div class="main-layout">
    
    <div class="selector-bar">
      <button class="preset-btn active" data-val="4">&radic;x = 4</button>
      <button class="preset-btn" data-val="5">&radic;x = 5</button>
      <button class="preset-btn" data-val="7">&radic;x = 7</button>
    </div>

    
    <div class="workspace">
      
      <div class="panel">
        <div class="algebra-flow">
          <div class="step-row" id="row-0">
            <span class="step-badge">Given</span>
            <span class="math-expr" id="expr-0"></span>
          </div>
          <div class="step-row" id="row-1">
            <span class="step-badge">Square ( )&sup2;</span>
            <span class="math-expr" id="expr-1"></span>
          </div>
          <div class="step-row" id="row-2">
            <span class="step-badge">Solve</span>
            <span class="math-expr" id="expr-2"></span>
          </div>
        </div>
      </div>

      
      <div class="panel">
        <div class="geo-container">
          <div class="square-box" id="geo-box">
            <span class="side-label-top" id="label-top"></span>
            <span class="side-label-left" id="label-left"></span>
            <div class="area-label" id="label-area"></div>
          </div>
        </div>
      </div>
    </div>

    
    <div class="controls">
      <button class="nav-btn" id="prev-btn" aria-label="Previous step">&#8592;</button>
      <div class="step-dots">
        <span class="dot active" data-step="0"></span>
        <span class="dot" data-step="1"></span>
        <span class="dot" data-step="2"></span>
      </div>
      <button class="nav-btn" id="next-btn" aria-label="Next step">&#8594;</button>
    </div>
  </div>

  <script>
    let k = 4;
    let step = 0;

    function renderMath() {
      const xVal = k * k;

      if (window.katex) {
        katex.render(`\\sqrt{x} = ${k}`, document.getElementById('expr-0'));
        katex.render(`(\\sqrt{x})^2 = ${k}^2`, document.getElementById('expr-1'));
        katex.render(`x = ${xVal}`, document.getElementById('expr-2'));
        
        katex.render(`\\sqrt{x} = ${k}`, document.getElementById('label-top'));
        katex.render(`\\sqrt{x} = ${k}`, document.getElementById('label-left'));

        if (step === 0) {
          katex.render(`\\text{Area } = x`, document.getElementById('label-area'));
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          katex.render(`x = ${k}^2`, document.getElementById('label-area'));
        } else {
          katex.render(`x = ${xVal}`, document.getElementById('label-area'));
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        document.getElementById('expr-0').textContent = `√x = ${k}`;
        document.getElementById('expr-1').textContent = `(√x)² = ${k}²`;
        document.getElementById('expr-2').textContent = `x = ${xVal}`;
        document.getElementById('label-top').textContent = `√x = ${k}`;
        document.getElementById('label-left').textContent = `√x = ${k}`;
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        k = parseInt(btn.getAttribute('data-val'), 10);
        step = 0;
        renderMath();
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    // Attach Nav Controls
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    document.getElementById('next-btn').addEventListener('click', () => setStep(step === 2 ? 0 : step + 1));

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      d.addEventListener('click', () => {
        setStep(parseInt(d.getAttribute('data-step'), 10));
      });
    });

    // Initial render
    window.addEventListener('load', renderMath);
    renderMath();
  </script>
</body>
</html>

This property also connects directly to **rational exponents**, where the radical $\sqrt{x}$ is equivalent to $x^{1/2}$.

Squaring uses the **Power of a Power rule**: $(x^{1/2})^2 = x^{(1/2 \cdot 2)} = x^1 = x$.

You can also understand this through **rational exponents**: the radical $\sqrt{x}$ is equivalent to $x^{1/2}$.

Squaring applies the **Power of a Power rule**, multiplying the exponents: $(x^{1/2})^2 = x^{(1/2 \cdot 2)} = x^1 = x$.

How do we use this rule to solve an equation when $x$ is trapped under a radical sign?

To solve an equation like $\sqrt{x} = 7$, apply the **Equality Property of Squaring**: if two quantities are equal, squaring both sides keeps them equal.

Square both sides to eliminate the radical: $(\sqrt{x})^2 = 7^2$, which simplifies directly to $x = 49$.

<!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">
  <link rel="stylesheet" href="{{VENDOR_CDN}}/katex.min.css">
  <script src="{{VENDOR_CDN}}/katex.min.js"></script>
  <style>
    * {
      box-sizing: border-box;
      margin: 0;
      padding: 0;
      font-family: -apple-system, BlinkMacSystemFont, "Segoe UI", Roboto, Helvetica, Arial, sans-serif;
      user-select: none;
    }
    body {
      background: transparent;
      color: #1e2945;
      padding: 12px;
      display: flex;
      flex-direction: column;
      align-items: center;
      justify-content: center;
      min-height: 100%;
    }
    .main-stage {
      width: 100%;
      max-width: 480px;
      display: flex;
      flex-direction: column;
      align-items: center;
      gap: 16px;
    }
    /* Step Tabs */
    .step-tabs {
      display: flex;
      gap: 8px;
      width: 100%;
      justify-content: center;
    }
    .tab-btn {
      flex: 1;
      max-width: 100px;
      padding: 8px 4px;
      background: #f1f5f9;
      border: 1px solid #e2e8f0;
      border-radius: 8px;
      font-size: 13px;
      font-weight: 600;
      color: #64748b;
      cursor: pointer;
      text-align: center;
      transition: all 0.2s ease;
    }
    .tab-btn.active {
      background: #2563eb;
      border-color: #2563eb;
      color: #ffffff;
      box-shadow: 0 2px 6px rgba(37, 99, 235, 0.25);
    }

    /* Equation Display Box */
    .display-area {
      width: 100%;
      height: 160px;
      background: #ffffff;
      border-radius: 12px;
      border: 1px solid #e2e8f0;
      box-shadow: 0 4px 12px rgba(0, 0, 0, 0.04);
      display: flex;
      flex-direction: column;
      align-items: center;
      justify-content: center;
      position: relative;
      overflow: hidden;
    }
    .equation-view {
      position: absolute;
      display: flex;
      flex-direction: column;
      align-items: center;
      justify-content: center;
      gap: 8px;
      opacity: 0;
      transform: translateY(8px);
      transition: opacity 0.3s ease, transform 0.3s ease;
      pointer-events: none;
    }
    .equation-view.active {
      opacity: 1;
      transform: translateY(0);
      pointer-events: auto;
    }
    .math-main {
      font-size: 26px;
      color: #1e293b;
    }
    .tag {
      font-size: 12px;
      font-weight: 700;
      text-transform: uppercase;
      letter-spacing: 0.5px;
      padding: 4px 10px;
      border-radius: 16px;
    }
    .tag-blue { background: #eff6ff; color: #2563eb; }
    .tag-amber { background: #fef3c7; color: #d97706; }
    .tag-emerald { background: #ecfdf5; color: #059669; }

    /* Action Balance Indicator */
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      justify-content: space-between;
      width: 100%;
      background: #f8fafc;
      border: 1px solid #e2e8f0;
      border-radius: 8px;
      padding: 10px 16px;
      font-size: 13px;
      color: #475569;
    }
    .op-tag {
      display: inline-flex;
      align-items: center;
      gap: 4px;
      font-weight: 600;
    }

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      display: flex;
      align-items: center;
      gap: 12px;
    }
    .ctrl-btn {
      background: #ffffff;
      border: 1px solid #cbd5e1;
      color: #334155;
      padding: 8px 14px;
      border-radius: 8px;
      font-size: 14px;
      font-weight: 600;
      cursor: pointer;
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      align-items: center;
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      color: #ffffff;
      min-width: 72px;
    }
    .ctrl-btn.play-btn:active {
      background: #1d4ed8;
    }
  </style>
</head>
<body>

  <div class="main-stage">
    
    <div class="step-tabs">
      <button class="tab-btn active" onclick="goToStep(0)">1. Given</button>
      <button class="tab-btn" onclick="goToStep(1)">2. Square</button>
      <button class="tab-btn" onclick="goToStep(2)">3. Solve</button>
      <button class="tab-btn" onclick="goToStep(3)">4. Check</button>
    </div>

    
    <div class="display-area">
      
      <div class="equation-view active" id="view-0">
        <span class="tag tag-blue">Original Radical</span>
        <div class="math-main" id="eq-0"></div>
      </div>
      
      <div class="equation-view" id="view-1">
        <span class="tag tag-amber">Square Both Sides</span>
        <div class="math-main" id="eq-1"></div>
      </div>
      
      <div class="equation-view" id="view-2">
        <span class="tag tag-emerald">Radical Eliminated</span>
        <div class="math-main" id="eq-2"></div>
      </div>
      
      <div class="equation-view" id="view-3">
        <span class="tag tag-emerald">True Statement ✓</span>
        <div class="math-main" id="eq-3"></div>
      </div>
    </div>

    
    <div class="balance-bar">
      <span class="op-tag" id="bar-left">Inverse: $(\sqrt{\Box})^2 = \Box$</span>
      <span class="op-tag" id="bar-right" style="color:#2563eb; font-weight:700;">Balance: Both Sides</span>
    </div>

    
    <div class="controls">
      <button class="ctrl-btn" onclick="prevStep()">←</button>
      <button class="ctrl-btn play-btn" id="playBtn" onclick="togglePlay()">⏸</button>
      <button class="ctrl-btn" onclick="nextStep()">→</button>
    </div>
  </div>

  <script>
    const equations = [
      "\\sqrt{x} = 7",
      "(\\sqrt{x})^{\\color{#d97706}{2}} = 7^{\\color{#d97706}{2}}",
      "x = 49",
      "\\sqrt{49} = 7 \\implies 7 = 7"
    ];

    equations.forEach((eq, i) => {
      katex.render(eq, document.getElementById('eq-' + i), {
        throwOnError: false,
        displayMode: true
      });
    });

    let current = 0;
    let playing = true;
    let timer = null;

    function renderStep() {
      document.querySelectorAll('.tab-btn').forEach((btn, i) => {
        btn.classList.toggle('active', i === current);
      });
      document.querySelectorAll('.equation-view').forEach((view, i) => {
        view.classList.toggle('active', i === current);
      });
    }

    function goToStep(index) {
      current = index;
      renderStep();
    }

    function nextStep() {
      current = (current + 1) % 4;
      renderStep();
    }

    function prevStep() {
      current = (current - 1 + 4) % 4;
      renderStep();
    }

    function togglePlay() {
      playing = !playing;
      document.getElementById('playBtn').textContent = playing ? '⏸' : '▶';
      if (playing) startLoop();
      else clearInterval(timer);
    }

    function startLoop() {
      clearInterval(timer);
      timer = setInterval(() => {
        nextStep();
      }, 3500);
    }

    startLoop();
  </script>
</body>
</html>

Always verify your answer by substituting it back into the original equation: $\sqrt{49} = 7$, which is a true statement.

What happens if a radical equation equals a negative number?

Consider the equation $\sqrt{x} = -4$. By definition, the **principal square root** (the $\sqrt{\phantom{x}}$ symbol) represents only the non-negative square root of a number.

Because the principal square root of a real number can never produce a negative output, $\sqrt{x} = -4$ has **no real solution**.

<!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">
<title>Extraneous Solutions in Square Root Equations</title>
<link rel="stylesheet" href="{{VENDOR_CDN}}/katex.min.css">
<script src="{{VENDOR_CDN}}/katex.min.js"></script>
<script src="{{VENDOR_CDN}}/classace-diagram.js"></script>
<style>
  * { box-sizing: border-box; margin: 0; padding: 0; }
  body {
    font-family: -apple-system, BlinkMacSystemFont, "Segoe UI", Roboto, sans-serif;
    color: #1e2945;
    background: transparent;
    padding: 12px;
    display: flex;
    flex-direction: column;
    align-items: center;
    justify-content: center;
    min-height: 100%;
  }

  .main-wrap {
    width: 100%;
    max-width: 520px;
    display: flex;
    flex-direction: column;
    gap: 14px;
  }

  /* Navigation Pills */
  .nav-bar {
    display: flex;
    justify-content: center;
    gap: 8px;
  }
  .nav-btn {
    background: #ffffff;
    border: 1px solid #e2e8f0;
    padding: 6px 14px;
    border-radius: 20px;
    font-size: 13px;
    font-weight: 600;
    color: #64748b;
    cursor: pointer;
    box-shadow: 0 1px 2px rgba(0,0,0,0.04);
  }
  .nav-btn.active {
    background: #ef4444;
    color: #ffffff;
    border-color: #ef4444;
  }

  /* Stage Window */
  .stage-container {
    position: relative;
    height: 250px;
    width: 100%;
    overflow: hidden;
  }

  .step-panel {
    position: absolute;
    inset: 0;
    display: flex;
    flex-direction: column;
    justify-content: center;
    align-items: center;
    gap: 12px;
    opacity: 0;
    pointer-events: none;
    transition: opacity 0.3s ease;
  }
  .step-panel.active {
    opacity: 1;
    pointer-events: auto;
  }

  /* Math Card */
  .math-card {
    background: #ffffff;
    border: 1px solid #e2e8f0;
    border-radius: 12px;
    padding: 16px 20px;
    box-shadow: 0 2px 8px rgba(0,0,0,0.04);
    display: flex;
    flex-direction: column;
    align-items: center;
    gap: 10px;
    width: 100%;
    max-width: 380px;
  }

  .math-row {
    display: flex;
    align-items: center;
    justify-content: center;
    gap: 10px;
    font-size: 20px;
  }

  .arrow-down {
    color: #94a3b8;
    font-size: 16px;
  }

  .badge {
    display: inline-flex;
    align-items: center;
    gap: 6px;
    padding: 4px 12px;
    border-radius: 999px;
    font-size: 13px;
    font-weight: 700;
    letter-spacing: 0.3px;
  }
  .badge-danger {
    background: #fef2f2;
    color: #ef4444;
    border: 1px solid #fecaca;
  }
  .badge-neutral {
    background: #f1f5f9;
    color: #475569;
    border: 1px solid #e2e8f0;
  }

  /* Visual Comparison Axis */
  .range-box {
    display: flex;
    align-items: center;
    justify-content: space-around;
    width: 100%;
    padding: 12px 16px;
    background: #f8fafc;
    border-radius: 8px;
    border: 1px dashed #cbd5e1;
  }

  .range-item {
    display: flex;
    flex-direction: column;
    align-items: center;
    gap: 4px;
  }

  .tag-pos {
    color: #059669;
    font-weight: 700;
    font-size: 13px;
  }
  .tag-neg {
    color: #ef4444;
    font-weight: 700;
    font-size: 13px;
  }

  .strike-box {
    position: relative;
    display: inline-block;
  }
  .strike-box::after {
    content: '';
    position: absolute;
    top: 50%;
    left: -4px;
    right: -4px;
    height: 3px;
    background: #ef4444;
    transform: rotate(-12deg);
    border-radius: 2px;
  }
</style>
</head>
<body>

<div class="main-wrap">
  <div class="nav-bar">
    <button class="nav-btn active" id="btn-0">1. Square</button>
    <button class="nav-btn" id="btn-1">2. Check</button>
    <button class="nav-btn" id="btn-2">3. Why?</button>
  </div>

  <div class="stage-container">
    
    <div class="step-panel active" id="step-0">
      <div class="math-card">
        <div class="math-row" id="eq1-orig"></div>
        <div class="arrow-down">↓ (square both sides)</div>
        <div class="math-row" id="eq1-sq"></div>
        <div class="arrow-down">↓</div>
        <div class="math-row" id="eq1-res"></div>
      </div>
      <div class="badge badge-neutral">Potential Solution: x = 16</div>
    </div>

    
    <div class="step-panel" id="step-1">
      <div class="math-card">
        <div class="math-row" id="eq2-sub"></div>
        <div class="arrow-down">↓</div>
        <div class="math-row">
          <span id="eq2-eval"></span>
          <span style="font-size: 22px; color: #ef4444; font-weight: bold;">≠</span>
          <span id="eq2-target"></span>
        </div>
      </div>
      <div class="badge badge-danger">✕ Extraneous Solution</div>
    </div>

    
    <div class="step-panel" id="step-2">
      <div class="math-card">
        <div class="range-box">
          <div class="range-item">
            <span id="eq3-left"></span>
            <span class="tag-pos">≥ 0 (Always)</span>
          </div>
          <span style="font-size: 22px; color: #ef4444; font-weight: bold;">≠</span>
          <div class="range-item">
            <span id="eq3-right"></span>
            <span class="tag-neg">&lt; 0 (Negative)</span>
          </div>
        </div>
      </div>
      <div class="badge badge-danger">✕ No Real Solution</div>
    </div>
  </div>
</div>

<script>
  // Render Math Expressions
  katex.render("\\sqrt{x} = -4", $('#eq1-orig'));
  katex.render("(\\sqrt{x})^2 = (-4)^2", $('#eq1-sq'));
  katex.render("x = 16", $('#eq1-res'));

  katex.render("\\sqrt{16} \\stackrel{?}{=} -4", $('#eq2-sub'));
  katex.render("4", $('#eq2-eval'));
  katex.render("-4", $('#eq2-target'));

  katex.render("\\sqrt{x}", $('#eq3-left'));
  katex.render("-4", $('#eq3-right'));

  // Step Switcher
  const buttons = [$('#btn-0'), $('#btn-1'), $('#btn-2')];
  const panels = [$('#step-0'), $('#step-1'), $('#step-2')];

  function showStep(idx) {
    buttons.forEach((b, i) => b.classList.toggle('active', i === idx));
    panels.forEach((p, i) => p.classList.toggle('active', i === idx));
  }

  buttons.forEach((btn, idx) => {
    onTap(btn, () => showStep(idx));
  });
</script>

</body>
</html>

If you blindly squared both sides, you would get $x = 16$, but checking $\sqrt{16} = 4 \neq -4$ reveals that this false answer is an **extraneous solution** (an invalid solution created by squaring).

Before squaring any radical equation, take a quick look at the other side — if it is negative, you can stop immediately because no real solution exists.

math

Solving One-Step Square Root Equations. Solve simple square root equations by isolating the radical and squaring both sides.

Solving Square-Root Equations: One Solution

Solve Radical Equations I

Intermediate Algebra 2e: Section 8.6 - Solve Radical Equations

When introducing one-step square root equations, ground student understanding in inverse operations: just as subtraction undoes addition and division undoes multiplication, squaring undoes a square root. Explicitly link this to standard HSN.RN.A.2 by writing $\sqrt{x}$ as $x^{1/2}$ and applying the exponent power rule $((x^{1/2})^2 = x^1 = x)$. Emphasize the 'Isolate First' rule before squaring, and embed checking for extraneous solutions as a standard requirement of the solving algorithm.

Solving One-Step Square Root Equations

Undoing the Square Root

Solving by Squaring Both Sides