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61 lines
2 KiB
HTML
61 lines
2 KiB
HTML
<div class="book-prose">
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<p>
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A quadratic equation in one unknown has the form \(ax^2 + bx + c = 0\), where
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\(a \neq 0\). You can solve these by factorisation, the quadratic
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formula, or by reading information from the graph of \(y = ax^2 + bx + c\).
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</p>
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<h2 id="discriminant" class="book-prose__heading">The discriminant</h2>
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<p>
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The <strong>discriminant</strong> \(\Delta\) (delta) is defined below. It tells you how many
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<em>real</em> roots the equation has without fully solving it.
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</p>
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<div class="book-callout book-callout--formula" role="note">
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<p class="book-callout__label">Definition</p>
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<p class="book-formula">\[ \Delta = b^2 - 4ac \]</p>
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</div>
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<h2 id="nature-of-roots" class="book-prose__heading">Nature of roots</h2>
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<div class="book-table-wrap">
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<table class="book-table">
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<thead>
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<tr>
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<th scope="col">Discriminant</th>
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<th scope="col">Nature of roots</th>
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<th scope="col">Graph of \(y = ax^2 + bx + c\)</th>
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</tr>
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</thead>
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<tbody>
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<tr>
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<td>\(\Delta > 0\)</td>
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<td>Two distinct real roots</td>
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<td>Parabola cuts the x-axis twice</td>
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</tr>
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<tr>
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<td>\(\Delta = 0\)</td>
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<td>One repeated real root (equal roots)</td>
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<td>Vertex touches the x-axis</td>
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</tr>
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<tr>
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<td>\(\Delta < 0\)</td>
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<td>No real roots</td>
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<td>Parabola lies entirely above or below the x-axis</td>
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</tr>
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</tbody>
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</table>
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</div>
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<p>
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When \(\Delta \geq 0\), the quadratic formula gives the roots:
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\[ x = \frac{-b \pm \sqrt{\Delta}}{2a} \]
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</p>
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<h2 id="try-it-yourself" class="book-prose__heading">Try it yourself</h2>
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<p>
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Use the explorer below to change \(a\), \(b\), and \(c\). Watch how \(\Delta\) and the
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parabola change, and read off the nature of the roots.
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</p>
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</div>
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<div class="math-widget-mount" data-widget="explorer" data-pagefind-ignore></div>
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