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  <title>Solving Quadratic Equation with Complex Roots</title>
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<body>

  
  <div class="eq-header">
    <div class="eq-title">Quadratic Equation with Complex Roots</div>
    <div class="eq-math"><span class="math-sym">x</span><sup>2</sup> − 4<span class="math-sym">x</span> + 13 = 0</div>
  </div>

  
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      <div class="tab-num">1</div>
      <div class="tab-title">Coefficients</div>
    </button>
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      <div class="tab-num">2</div>
      <div class="tab-title">Discriminant</div>
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      <div class="tab-num">3</div>
      <div class="tab-title">Complex Root</div>
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      <div class="tab-num">4</div>
      <div class="tab-title">Conjugate Pair</div>
    </button>
  </div>

  
  <div class="detail-card" id="detail-card">
    <div class="detail-header">
      <div class="detail-title" id="detail-step-title">
        <span>Step 1: Identify Coefficients</span>
      </div>
      <div class="detail-badge" id="detail-step-badge">Standard Form</div>
    </div>
    <div class="detail-body" id="detail-content">
      
    </div>
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      <div class="step-row-left">
        <div class="step-row-num">1</div>
        <div class="step-row-name">Coefficients</div>
      </div>
      <div class="step-row-math"><span class="math-sym">a</span> = 1, <span class="math-sym">b</span> = −4, <span class="math-sym">c</span> = 13</div>
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      <div class="step-row-left">
        <div class="step-row-num">2</div>
        <div class="step-row-name">Discriminant Δ</div>
      </div>
      <div class="step-row-math">Δ = (−4)<sup>2</sup> − 4(1)(13) = <strong>−36</strong></div>
    </div>

    <div class="step-row" id="row-2" onclick="goToStep(2)">
      <div class="step-row-left">
        <div class="step-row-num">3</div>
        <div class="step-row-name">Square Root</div>
      </div>
      <div class="step-row-math">√<span style="text-decoration:overline">−36</span> = <strong>6<span class="math-sym">i</span></strong></div>
    </div>

    <div class="step-row" id="row-3" onclick="goToStep(3)">
      <div class="step-row-left">
        <div class="step-row-num">4</div>
        <div class="step-row-name">Conjugate Pair</div>
      </div>
      <div class="step-row-math">
        <div class="badge-pair">
          <span class="badge badge-pos">2 + 3<span class="math-sym">i</span></span>
          <span class="badge badge-neg">2 − 3<span class="math-sym">i</span></span>
        </div>
      </div>
    </div>
  </div>

  
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          <div class="detail-desc">Extract the values of <em>a</em>, <em>b</em>, and <em>c</em> from the quadratic equation.</div>
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              <span class="frac-top">4 ± 6<span class="math-sym">i</span></span>
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The Fundamental Theorem of Algebra and Quadratic Equations

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      align-items: center;
      justify-content: center;
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</head>
<body>

<div class="diagram-content">
  
  <div class="eq-section">
    <div class="eq-header">
      <span class="coef-a">2</span>x² + <span class="coef-b">3</span>x + <span class="coef-c">5</span> = 0
    </div>

    <div class="math-row">
      <span>Δ = 3² − 4(2)(5) = <strong>−31</strong></span>
      <span class="disc-tag">Δ &lt; 0 → Complex Conjugates</span>
    </div>

    <div class="math-row">
      <span id="pillReal" class="pill pill-re" onclick="setMode('real')" title="Click to highlight Real part">
        Re(z) = −0.75
      </span>
      <span id="pillImag" class="pill pill-im" onclick="setMode('imag')" title="Click to highlight Imaginary part">
        Im(z) = ±1.39 i
      </span>
    </div>
  </div>

  
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<body>

  <div class="diagram-grid">
    
    <div class="panel">
      <div class="panel-title">Real Plane</div>
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      <div class="badge badge-red">0 Real Roots (x² = -1)</div>
    </div>

    
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      <div class="panel-title">Complex Plane</div>
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      <div class="badge badge-cyan">2 Complex Roots</div>
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  </div>

</body>
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  <meta charset="UTF-8">
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  <title>Fundamental Theorem of Algebra</title>
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<body>

  
  <div class="banner">
    <span>Degree <span class="highlight">n</span> Polynomial</span>
    <span>➔</span>
    <span>Exactly <span class="highlight">n</span> Complex Roots</span>
  </div>

  
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    <div class="poly-card active" onclick="selectCard(0)" id="card-0">
      <div class="degree-tag">Degree 1</div>
      <div class="card-title">Linear</div>
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      <div class="eqn">ax + b</div>
      <div class="roots-badge">● 1 Root</div>
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      <div class="degree-tag">Degree 2</div>
      <div class="card-title">Quadratic</div>
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      <div class="eqn">ax² + bx + c</div>
      <div class="roots-badge">●● 2 Roots</div>
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      <div class="degree-tag">Degree 3</div>
      <div class="card-title">Cubic</div>
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      <div class="eqn">ax³ + bx² + ...</div>
      <div class="roots-badge">●●● 3 Roots</div>
    </div>

  </div>

  
  <div class="detail-strip">
    <div class="detail-text" id="detailText">
      <span>Highest power = <span class="num">1</span></span>
      <span>➔</span>
      <span>Always <span class="num">1</span> root (real or complex)</span>
    </div>
    <div class="nav-row">
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      <button class="step-btn" onclick="nextCard()">▶</button>
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    const details = [
      '<span>Highest power = <span class="num">1</span></span> <span>➔</span> <span>Always <span class="num">1</span> root (real or complex)</span>',
      '<span>Highest power = <span class="num">2</span></span> <span>➔</span> <span>Always <span class="num">2</span> roots (real or complex)</span>',
      '<span>Highest power = <span class="num">3</span></span> <span>➔</span> <span>Always <span class="num">3</span> roots (real or complex)</span>'
    ];

    function selectCard(index) {
      currentIndex = index;
      for (let i = 0; i < 3; i++) {
        const card = document.getElementById('card-' + i);
        const dot = document.getElementById('dot-' + i);
        if (i === index) {
          card.classList.add('active');
          dot.classList.add('active');
        } else {
          card.classList.remove('active');
          dot.classList.remove('active');
        }
      }
      document.getElementById('detailText').innerHTML = details[index];
    }

    function nextCard() {
      selectCard((currentIndex + 1) % 3);
    }

    function prevCard() {
      selectCard((currentIndex - 1 + 3) % 3);
    }
  </script>
</body>
</html>

OGM.12.NUM.4

Aligned learning resources

Related diagrams