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<title>Algebra Mistakes</title>
<meta name="description" content="We have gathered here a collection of mistakes that are pretty easy to make in Algebra." />
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<h1 class="center">Algebra Mistakes</h1>
<p>We have gathered here a collection of mistakes that are pretty easy to make.</p>
<p>Try to avoid these!</p>
<div class="simple">
<table border="0" align="center">
<tr align="center">
<th width="50%">Mistake
<br><img src="../images/style/no.svg" height="46" alt="not" /> </th>
<th width="50%">Correction
<br><img src="../images/style/yes.svg" alt="yes" height="46" /></th>
</tr>
<tr align="center">
<td>x<sup>2</sup> = 25, so x = 5</td>
<td>x = 5 <span class="hilite">or x = &minus;5</span></td>
</tr>
<tr align="center">
<td>(x&minus;5)<sup>2</sup> = x<sup>2</sup> &minus; 25</td>
<td>= <span class="hilite">(x&minus;5)(x&minus;5)</span> = x<sup>2</sup> &minus; 10x + 25</td>
</tr>
<tr align="center">
<td>&radic;(x<sup>2</sup>+y<sup>2</sup>) = x + y</td>
<td>&radic;(x<sup>2</sup>+y<sup>2</sup>) is as far as we can go</td>
</tr>
<tr align="center">
<td>&nbsp;</td>
<td>&nbsp;</td>
</tr>
<tr align="center">
<td>x<sup>2</sup>x<sup>4 </sup>= x<sup>8</sup></td>
<td>= x<sup class="hilite">6</sup> (add exponents)</td>
</tr>
<tr align="center">
<td>(x<sup>2</sup>)<sup>4 </sup>= x<sup>6</sup></td>
<td>= x<sup class="hilite">8</sup> (multiply exponents)</td>
</tr>
<tr align="center">
<td>2x<sup>-1</sup> = 1/(2x)</td>
<td>= <span class="hilite">2/x</span></td>
</tr>
<tr align="center">
<td>&minus;5<sup>2</sup> = 25</td>
<td>= <span class="hilite">&minus;</span>25 (do <a href="../operation-order-pemdas.html">exponent before minus</a>)</td>
</tr>
<tr align="center">
<td>(&minus;5)<sup>2</sup> = &minus;25</td>
<td>= <span class="hilite">+</span>25 (do <a href="../operation-order-pemdas.html">brackets before exponent</a>)</td>
</tr>
<tr align="center">
<td>5<sup>&frac12;</sup> = 1/5<sup>2</sup></td>
<td>= <span class="hilite">&radic;5</span></td>
</tr>
<tr align="center">
<td>&nbsp;</td>
<td>&nbsp;</td>
</tr>
<tr align="center">
<td>log(a+b) = log(a) + log(b)</td>
<td>log(a+b) is as far as we can go</td>
</tr>
<tr align="center">
<td>&nbsp;</td>
<td>&nbsp;</td>
</tr>
<tr align="center">
<td>x(a/b) = xa/xb</td>
<td>= <span class="hilite">xa</span>/b</td>
</tr>
<tr align="center">
<td>x&minus;(5+a) = x&minus;5+a</td>
<td> = x&minus;5<span class="hilite">&minus;</span>a</td>
</tr>
<tr align="center">
<td>&nbsp;</td>
<td>&nbsp;</td>
</tr>
<tr align="center">
<th><img src="../images/style/no.svg" height="46" alt="not" /> </th>
<th><img src="../images/style/yes.svg" alt="yes" height="46" /></th>
</tr>
</table>
</div>
<p>And be careful of these ones too:</p>
<h2>Simplifying Fractions</h2>
<table border="0" align="center">
<tr>
<td class="larger"><span class="intbl"><em>x</em><strong>x+y</strong></span> = <span class="intbl"><em>x</em><strong>x</strong></span> + <span class="intbl"><em>x</em><strong>y</strong></span></td>
<td>&nbsp;</td>
<td><img src="../images/style/no.svg" height="46" alt="not" /></td>
</tr>
</table>
<p>We can't simplify that!</p>
<p>Imagine x=4 and y=5:</p>
<p class="center large"><span class="intbl"><em>4</em><strong>4+5</strong></span> = <span class="intbl"><em>4</em><strong>9</strong></span></p>
<p>That is definitely not equal to <span class="intbl"><em>4</em><strong>4</strong></span> + <span class="intbl"><em>4</em><strong>5</strong></span> (which actually equals more than 1)</p>
<p>Maybe you were thinking of this kind of fraction that we <b>can</b> simplify:</p>
<table border="0" align="center">
<tr>
<td class="larger"><span class="intbl"><em>x+y</em><strong>x</strong></span> = <span class="intbl"><em>x</em><strong>x</strong></span> + <span class="intbl"><em>y</em><strong>x</strong></span></td>
<td>&nbsp;</td>
<td><img src="../images/style/yes.svg" alt="yes" height="46" /></td>
</tr>
</table>
<h2>Square root of xy</h2>
<p class="center"><span class="larger"><b></b></span></p>
<p><span class="larger"><b>&radic;(xy) =&radic;x&radic;y</b><i> </i></span>... but not always!</p>
<div class="example">
<h3>Example: x = &minus;5 and y = &minus;2</h3>
<div class="tbl">
<div class="row"><span class="left">&radic;10 =</span><span class="right"> &radic;(&minus;5 &times; &minus;2)</span></div>
<div class="row"><span class="left">&nbsp;=</span><span class="right">&radic;(&minus;5)&radic;(&minus;2) &nbsp; <i>(the mistake)</i></span></div>
<div class="row"><span class="left">&nbsp;=</span><span class="right">i&radic;5 &times; i&radic;2</span></div>
<div class="row"><span class="left">&nbsp;=</span><span class="right">i<sup>2</sup>&radic;5&radic;2</span></div>
<div class="row"><span class="left">&nbsp;=</span><span class="right">&minus;&radic;10</span></div>
</div>
<p>So, does <b>&radic;10</b> = <b>&minus;&radic;10</b> ??? I think not!</p>
</div>
<div class="center80">
<p class="center"><span class="larger"><b>&radic;(xy) =&radic;x&radic;y</b> <i>only when <b>x</b> and <b>y</b> are both &gt;= 0</i></span></p>
</div>
<h2>Two Equals One</h2>
<div class="example">
<h3>Example:</h3>
<div class="tbl">
<div class="row"><span class="left">Start with:</span><span class="right"> x = y </span></div>
<div class="row"><span class="left">Multiply both sides by x:</span><span class="right"> x<sup>2</sup> = xy </span></div>
<div class="row"><span class="left">Subtract y<sup>2</sup> from both sides:</span><span class="right"> x<sup>2</sup> &minus; y<sup>2</sup> = xy &minus; y<sup>2</sup> </span></div>
<div class="row"><span class="left">Factor:</span><span class="right">(x+y)(x&minus;y) = y(x&minus;y)</span></div>
<div class="row"><span class="left">Divide both sides by (x&minus;y):</span><span class="right">x + y = y &nbsp; <i>(the mistake)</i> </span></div>
<div class="row"><span class="left">Since x = y, we see that:</span><span class="right"> 2y = y </span></div>
<div class="row"><span class="left">And so:</span><span class="right"> 2 = 1 </span></div>
</div>
<p><b>Hang on! That can't be right!</b></p>
</div>
<p>What went wrong? Silly us! We tried to <a href="../numbers/dividing-by-zero.html">divide by zero</a>.</p>
<p>When we said that x=y, it means that <b>(x&minus;y)=0</b> , so going from <b>(x+y)(x&minus;y) = y(x&minus;y)</b> to <b>x + y = y</b> is a mistake.</p>
<h2>Factoring</h2>
<div class="example">
<h3>Example: Solve x<sup>2</sup> – 5x = 2</h3>
<div class="tbl">
<div class="row"><span class="left">Start with:</span><span class="right">x<sup>2</sup> – 5x = 2 </span></div>
<div class="row"><span class="left">Factor x:</span><span class="right">x(x&minus;5) = 2 </span></div>
<div class="row"><span class="left">So:</span><span class="right">x=2 or x&minus;5=2 &nbsp; <i>(the mistake)</i></span></div>
<div class="row"><span class="left">And so:</span><span class="right">x=2 or 7</span></div>
</div>
<p>Let's check&nbsp;x=2:</p>
<p class="center"><span class="right">2<sup>2</sup> – 5&times;2 = 4&minus;10 = <b>&minus;6</b>, but we wanted </span><span class="right">x<sup>2</sup> – 5x = <b>2</b></span></p>
<p>That only works when <b>x(x&minus;5) =<span class="hilite"> 0 </span></b>(zero) not any other number</p>
</div>
<p>&nbsp;</p>
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<a href="../operation-order-pemdas.html">Order of Operations</a>
<a href="index.html">Algebra Index</a>
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