<p>For the curve \(y=ax^2+bx+c\), \(a\ne0\):</p>
<div class="content-box-placement content-box-placement--full content-box-placement--align-center" style="display:flex;justify-content:stretch;max-width:100%"><div class="content-box content-box--emphasis" data-content-box-preset="emphasis" style="box-sizing:border-box;border-color:var(--content-box-border, #1E6FBA);border-style:solid;border-width:0px;border-radius:0px;background-color:var(--content-box-background, #F5F8FC);color:var(--content-box-foreground, #111827);padding:12px 16px;font-family:'EB Garamond', serif;font-size:16px;line-height:1.35;width:100%;max-width:100%;border-left-width:3px"><p style="text-align: center">Discriminant, \(D=b^2-4ac\)</p></div></div>
<p class="paragraph-header">Nature of the roots</p>
<div class="content-block content-block--table"><div class="table-container"><table style="table-layout: fixed; width: 580px;"><colgroup><col style="width: 19.8276%;"><col style="width: 25.0000%;"><col style="width: 27.5862%;"><col style="width: 27.5862%;"></colgroup><thead><tr><th style="text-align: center; --cell-justify: center"><strong>Discriminant</strong></th><th style="text-align: center; --cell-justify: center"><strong>Nature of roots</strong></th><th style="text-align: center; --cell-justify: center"><strong>\(a&gt;0\)</strong></th><th style="text-align: center; --cell-justify: center"><strong>\(a&lt;0\)</strong></th></tr></thead><tbody><tr><td style="text-align: center; --cell-justify: center">\(b^2-4ac&gt;0\)</td><td class="table-cell--fill-in" style="text-align: center; --cell-justify: center"></td><td style="text-align: center; --cell-justify: center"><div class="graph-block" data-graph-config="eyJheGlzIjp0cnVlLCJncmlkIjpmYWxzZSwid2lkdGgiOjE0MCwiaGVpZ2h0IjoxMDAsImVsZW1lbnRzIjpbeyJpZCI6InUwMS10MS1jZzEtZiIsInR5cGUiOiJmdW5jdGlvbmdyYXBoIiwiY29sb3IiOiIjMDAwMDAwIiwieE9mZnNldCI6MCwieU9mZnNldCI6MCwiZXhwcmVzc2lvbiI6InheMi00Iiwic3Ryb2tlV2lkdGgiOjEuNX0seyJ4IjotMiwieSI6MCwiaWQiOiJ1MDEtdDEtY2cxLXAxIiwidHlwZSI6InBvaW50IiwiY29sb3IiOiIjMDAwMDAwIiwiZml4ZWQiOnRydWUsImxhYmVsIjoiIiwicG9pbnRTaXplIjoyLCJsYWJlbE9mZnNldCI6WzAsMF19LHsieCI6MiwieSI6MCwiaWQiOiJ1MDEtdDEtY2cxLXAyIiwidHlwZSI6InBvaW50IiwiY29sb3IiOiIjMDAwMDAwIiwiZml4ZWQiOnRydWUsImxhYmVsIjoiIiwicG9pbnRTaXplIjoyLCJsYWJlbE9mZnNldCI6WzAsMF19XSwiYWxpZ25tZW50IjoiY2VudGVyIiwic2hvd1hBeGlzIjp0cnVlLCJzaG93WUF4aXMiOnRydWUsInhBeGlzTGFiZWwiOiJcXCh4XFwpIiwieUF4aXNMYWJlbCI6IlxcKHlcXCkiLCJib3VuZGluZ0JveCI6Wy0zLjUsNC41LDMuNSwtNS41XSwib3JpZ2luTGFiZWwiOiJcXChPXFwpIiwic2hvd0F4aXNMYWJlbHMiOmZhbHNlLCJzaG93TmF2aWdhdGlvbiI6ZmFsc2UsImtlZXBBc3BlY3RSYXRpbyI6ZmFsc2UsInhBeGlzTGFiZWxPZmZzZXQiOlstNSwtMTNdLCJ5QXhpc0xhYmVsT2Zmc2V0IjpbLTEzLC0xXSwib3JpZ2luTGFiZWxPZmZzZXQiOlstMTAsLTEyXX0=" data-width="140" data-height="100" data-proportional style="width:140px;height:100px;max-width:100%;display:flow-root;margin:0 auto;--graph-width:140px;--graph-aspect:140 / 100;"></div></td><td style="text-align: center; --cell-justify: center"><div class="graph-block" data-graph-config="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" data-width="140" data-height="100" data-proportional style="width:140px;height:100px;max-width:100%;display:flow-root;margin:0 auto;--graph-width:140px;--graph-aspect:140 / 100;"></div></td></tr><tr><td style="text-align: center; --cell-justify: center">\(b^2-4ac=0\)</td><td class="table-cell--fill-in" style="text-align: center; --cell-justify: center"></td><td style="text-align: center; --cell-justify: center"><div class="graph-block" data-graph-config="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" data-width="140" data-height="100" data-proportional style="width:140px;height:100px;max-width:100%;display:flow-root;margin:0 auto;--graph-width:140px;--graph-aspect:140 / 100;"></div></td><td style="text-align: center; --cell-justify: center"><div class="graph-block" data-graph-config="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" data-width="140" data-height="100" data-proportional style="width:140px;height:100px;max-width:100%;display:flow-root;margin:0 auto;--graph-width:140px;--graph-aspect:140 / 100;"></div></td></tr><tr><td style="text-align: center; --cell-justify: center">\(b^2-4ac&lt;0\)</td><td class="table-cell--fill-in" style="text-align: center; --cell-justify: center"></td><td style="text-align: center; --cell-justify: center"><div class="graph-block" data-graph-config="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" data-width="140" data-height="100" data-proportional style="width:140px;height:100px;max-width:100%;display:flow-root;margin:0 auto;--graph-width:140px;--graph-aspect:140 / 100;"></div></td><td style="text-align: center; --cell-justify: center"><div class="graph-block" data-graph-config="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" data-width="140" data-height="100" data-proportional style="width:140px;height:100px;max-width:100%;display:flow-root;margin:0 auto;--graph-width:140px;--graph-aspect:140 / 100;"></div></td></tr><tr><td style="text-align: center; --cell-justify: center">\(b^2-4ac\ge 0\)</td><td class="table-cell--fill-in" style="text-align: center; --cell-justify: center"></td><td class="table-cell--fill-in" style="text-align: center; --cell-justify: center"></td><td class="table-cell--fill-in" style="text-align: center; --cell-justify: center"></td></tr></tbody></table></div></div>
<p>The sign of \(a\) determines whether the parabola opens upwards or downwards. The discriminant conditions are the same.</p>
<p>Write the equation with zero on one side before identifying \(a,b,c\).</p>

For the curve \(y=ax^2+bx+c\), \(a\ne0\):

Two real and distinct roots. The curve cuts the \(x\)-axis at two points.

Two real and equal roots. The curve touches the \(x\)-axis; the \(x\)-axis is a tangent to the curve.

No real roots. The curve does not meet the \(x\)-axis.

The curve meets the \(x\)-axis at one or two points.

The sign of \(a\) determines whether the parabola opens upwards or downwards. The discriminant conditions are the same.

Discriminant	Nature of roots	\(a>0\)	\(a<0\)
\(b^2-4ac>0\)
\(b^2-4ac=0\)
\(b^2-4ac<0\)
\(b^2-4ac\ge 0\)