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      <text>7.1.6 (Intersections with quadratics)</text>
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      <text><![CDATA[<h1>Intersections with quadratics</h1>


<p>We spoke about the intersections between straight lines and quadratics in week 1. We return to the topic here to solve more simultaneous equations. Recall that the graph of the quadratic \(ax^2+bx+c\) consists of all points \((x,y)\) which satisfy \(y=ax^2+bx+c\). So:</p>
<p> </p>
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<p>if a point \((x,y)\) is on the parabola then we know \(y=ax^2+bx+c\)</p>
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<p>for points not on the parabola, we know \(y\neq ax^2+bx+c\).</p>
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<p>If we want to find the points of intersection of the line \(y=mx+k\) with the parabola \(y=ax^2+bx+c\), we solve \[ mx+k = ax^2+bx+c. \quad\quad(\ast) \] </p>
<p>Another way to think about this is to measure the vertical distance between the line and parabola at a given \(x\) value, which is given by</p>
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<td>\[ \underbrace{\left(ax^2+bx+c\right)}_{\substack{\text{height of point} \\ \text{on parabola}}} - \underbrace{\left(mx+k\right)}_{\substack{\text{height of point} \\ \text{on line}}} \]</td>
<td><img class="img-responsive atto_image_button_text-top" src="@@PLUGINFILE@@/3_2_1_distance.svg" alt="Distance between line and parabola" width="141" height="103" /></td>
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<p> </p>
<p>This shows that finding roots of \((\ast)\) is equivalent to finding \(x\)-coordinates where there is no distance between the points on the line and parabola.</p>
<p>Now, \((\ast)\) is a quadratic (we can write it in standard form if we wished by rearranging) so its discriminant, \(\Delta\), tells us about its roots:</p>
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<tr style="background: #c1ddea;"><th scope="col">Discriminant</th><th scope="col">Interpretation</th><th scope="col">Sketch</th></tr>
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<td>\(\Delta&gt;0\)</td>
<td>there are two distinct roots, so the line and curve intersect twice</td>
<td><img class="img-responsive atto_image_button_middle" src="@@PLUGINFILE@@/3_2_1_pos.svg" alt="Parabola and line with two intersections" width="64" height="52" /></td>
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<td>\(\Delta=0\)</td>
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<p align="left">there is one real root, so the line and curve intersect once</p>
(i.e. the line is a tangent to the curve)</td>
<td><img class="img-responsive atto_image_button_middle" src="@@PLUGINFILE@@/3_2_1_zero.svg" alt="Parabola and line with one intersection" width="64" height="62" /></td>
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<td>\(\Delta&lt;0\)</td>
<td>there are no real roots, so the line and the parabola do not intersect</td>
<td><img class="img-responsive atto_image_button_middle" src="@@PLUGINFILE@@/3_2_1_neg.svg" alt="Parabola and line with no intersections" width="64" height="68" /></td>
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</table>]]></text>
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