7. Incomplete quadratic equations IIhttps://peda.net/id/29e2dc512cf2022-09-05T12:42:41+03:00https://peda.net/:static/533/peda.net.logo.bg.svg<div class="license">Tämän sivun lisenssi <a rel="license" href="https://peda.net/info">Peda.net-yleislisenssi</a></div>
Incomplete quadratic equations IIhttps://peda.net/id/29e91a642cf2020-10-27T13:49:42+02:00<p class="p1">Let's draw two incomplete quadratic polynomial functions of the form [[$ f (x) = x^2 + bx $]] in the same coordinate system.</p>
<a href="https://peda.net/qis/2022-2023/mathematics/maths-grade-7/oikjs/7vtayi/images/k7vtayi/kuva-1#top" title="Skärmavbild 2018-12-23 kl. 22.38.56.png"><img src="https://peda.net/qis/2022-2023/mathematics/maths-grade-7/oikjs/7vtayi/images/k7vtayi/kuva-1:file/photo/eaa40c8714510d0c1bec42f06978cc73e72c1880/Ska%CC%88rmavbild%202018-12-23%20kl.%2022.38.56.png" alt="" title="Congruent incomplete quadratic functions" class="inline" loading="lazy"/></a> <br/>
<p class="p1">The graphs of the functions are <b>congruent</b>. The coefficient of the first-order term [[$ b $]] is clearly related to the second zero of the function and the position of the parabola's axis of symmetry.</p>
<br/>
<div class="eoppi-summary"><b>The answers for equation[[$ ax^2 + bx = 0 $]].</b><br/>
<ul>
<li>There are always two solutions to the equation.</li>
<li>One of the solutions is always x = 0.</li>
</ul>
</div>
<p class="p1">Equations of the form [[$ y = ax^2 + bx $]] are most conveniently solved by dividing them into factors.</p>
<div class="eoppi-summary"><b>Solving an equation of the form [[$ ax^2 + bx = 0 $]]:</b><br/>
<ul>
<li>Divide the equation into the factors so that both factors contain the variable [[$ x $]].</li>
<li>The <b>zero-product property</b> applies. The product is only zero if any of the <b>factors</b> is zero.</li>
</ul>
</div>
<h3><b>Example 1</b></h3>
<p class="p1">Solve the equation [[$ 2x^2 + 6x = 0 $]] and sketch the graph of the parabola determined by the equation.<br/>
<br/>
[[$ \begin{align*}
2x^2 + 6x &= 0 \;\;\; {\color {red} {\text {Divide the equation by factors.}}} \\
2x(x + 3) &= 0 \\
\end{align*} $]]</p>
<p class="p1">Apply the zero-product property.<br/>
<br/>
[[$ \begin{align*}
2x &= 0 \;\;\;\;\;\;\;\;\;\;\;\;\;\;\; {\text {or}}\;\;\;\;\;\; \;\;\;\;\;\; x + 3 &= 0 \\
x &= 0 \;\;\;\;\;\; \;\;\;\;\;\; \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; \;\;\;\;\;\; x &= -3 \\
\end{align*} $]]<br/>
<a href="https://peda.net/qis/2022-2023/mathematics/maths-grade-7/oikjs/7vtayi/images/k7vtayi/kuva-2#top" title="Skärmavbild 2018-12-23 kl. 22.56.55.png"><img src="https://peda.net/qis/2022-2023/mathematics/maths-grade-7/oikjs/7vtayi/images/k7vtayi/kuva-2:file/photo/5a7837fd5b223b9795265ac9764eaf5514edc08c/Ska%CC%88rmavbild%202018-12-23%20kl.%2022.56.55.png" alt="" title="Example 1" class="inline" loading="lazy"/></a><br/>
<b>Answer:</b> <em></em>[[$ x = 0 $]] or <em></em>[[$ x = -3 $]]</p>
<h3><b>Example 2</b></h3>
<p class="p1">Solve the equation [[$ 5x^2 - 2x = 0 $]] and sketch the graph of the parabola determined by the equation.<br/>
<br/>
[[$ \begin{align*}
5x^2 - 2x &= 0 \;\;\; {\color {red} {\text {Divide the equation by factors.}}} \\
x(5x - 2) &= 0 \\
\end{align*} $]]</p>
<p class="p1">Apply the zero-product property.</p>
[[$ \begin{align*}
x &= 0 \;\;\;\;\;\;\;\;\;\;\;\;\;\;\; {\text {or}}\;\;\;\;\;\; \;\;\;\;\;\; 5x - 2 &= 0 \\
x &= 0 \;\;\;\;\;\; \;\;\;\;\;\; \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; \;\;\;\;\;\; 5x &= 2 \;\;\; {\color {blue} {||:5}} \\
x &&= \displaystyle\frac {2} {5}
\end{align*} $]] <br/>
<br/>
<a href="https://peda.net/qis/2022-2023/mathematics/maths-grade-7/oikjs/7vtayi/images/k7vtayi/kuva-3#top" title="Skärmavbild 2018-12-23 kl. 23.01.44.png"><img src="https://peda.net/qis/2022-2023/mathematics/maths-grade-7/oikjs/7vtayi/images/k7vtayi/kuva-3:file/photo/df5cab5a886084a7fb2acd2d6678f4c222abd111/Ska%CC%88rmavbild%202018-12-23%20kl.%2023.01.44.png" alt="" title="Example 2" class="inline" loading="lazy"/></a> <br/>
<p class="p1"><b>Answer:</b> <em></em>[[$ x = 0 $]] or [[$ x = \displaystyle\frac {2} {5} $]]</p>
2022-09-05T12:42:41+03:00Exerciseshttps://peda.net/id/29e9b83f2cf2020-10-14T09:28:05+03:00<a href="https://peda.net/id/29ea4f262cf" rel="noopener" target="_blank">Basic exercises<br/>
</a><a href="https://peda.net/id/29f01acf2cf" rel="noopener" target="_blank">Applied exercises<br/>
</a><a href="https://peda.net/id/29f771f32cf" rel="noopener" target="_blank">Challenging exercises<br/>
</a>2022-09-05T12:42:41+03:00