Common Algebraic Errors

Eleanor Roosevelt said, “Learn from the mistakes of others. You can’t live long enough to make them all yourself.” When it comes to algebra, there are a lot of common mathematical errors that can accidentally be done during either an exam or just when studying, learning, or just trying to improve your mathematical skills. Lindsay Fox said it clearly when she quoted that “Learning by making mistakes and not duplicating them is what life is about.” Some mistakes are created and/or caused by whether being in a hurry or even while being lazy to even just being overconfident. Anyways, to improve on those mistake, this is dedicated to detailing the common algebraic errors and what is the right way to solve these problems.

\[\begin{matrix} \frac{2}{0}\ne 0&\textup{and}&\frac{2}{0}\ne 2 \end{matrix}\]

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So, what make this an error? Well, when you divide anything by zero, it is just going to be undefined. Think about it: What happens when you divide something by nothing?

\[-3^2=9\]

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So, what make this an error? Well, when it comes to dealing with exponents, the exponent affects that it's next no and not the whole term. For example, the exponent in \(-3^2\) affects only the \(3\) and not the \(-3\) since \[\begin{matrix} -3^2&&(-3)^2\\ =-(3\cdot 3)&&=(-3)\cdot (-3)\\ =-(9)&&=(-)(-)(3)(3)\\ =-9&&=9\\ &-9\neq 9& \end{matrix}\]

\[(x^2)^3\neq x^5\]

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So, what make this an error? Well, when it comes to dealing with exponents, \(x^2)^3\) means that you are multiplying \(x^2)\) 3 times, which means \[\begin{matrix} (x^2)^3&=(x^2)(x^2)(x^2)\\ &=x^{2+2+2}\\ &=x^{6} \end{matrix}\]

\[\frac{a}{b+c}\ne \frac{a}{b}+\frac{a}{c}\]

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So, what make this an error? Well, when it comes to dealing with fractions, the denominator can not be separated to be simplified. Here is a good example: \[\frac{1}{2}=\frac{1}{1+1}\ne \frac{1}{1}+\frac{1}{1}=2\]

\[\frac{1}{x^2+x^3}=x^{-2}+x^{-3}\]

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So, what make this an error? Well, when it comes to dealing with fractions, the denominator can not be separated to be simplified even if it is already simplified. Here is a good example: \[\frac{1}{x^2+x^3}=\frac{1}{x^2\left(x+1\right)}=x^{-2}\left(x+1\right)^{-1}\]

\[\frac{a+bx}{a}\ne 1+bx\]

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So, what make this an error? Well, when it comes to dealing with fractions, you can't simplify one part of the fraction; you have to make sure that all parts are simplified. Here is a good example: \[\frac{a+bx}{a}=\frac{a}{a}+\frac{bx}{a}=1+\frac{bx}{a}\]

\[\frac{a}{\left(\frac{b}{c}\right)}\ne \frac{ab}{c}\]

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So, what make this an error? Well, when it comes to dealing with fractions, you must make sure that you recall that \(a=\frac{a}{1}\) then understand that a fraction as a denominator means to multiply its reciprocal by the numerator. Here is a good example: \[\frac{a}{\left(\frac{b}{c}\right)}=\frac{\left(\frac{a}{1}\right)}{\left(\frac{b}{c}\right)}=\frac{a}{1}\div \frac{b}{c}=\frac{a}{1}\cdot \frac{c}{b}=\frac{ac}{b} \]

\[\frac{\left(\frac{a}{b}\right)}{c}\ne \frac{ab}{c}\]

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So, what make this an error? Well, when it comes to dealing with fractions, you must make sure that you recall that \(c=\frac{c}{1}\) then understand that a fraction as a numerator means to multiply its reciprocal by the denominator. Here is a good example: \[\frac{\left(\frac{a}{b}\right)}{c}=\frac{\left(\frac{a}{b}\right)}{\left(\frac{c}{1}\right)}=\frac{a}{b}\div \frac{c}{1}=\frac{a}{b}\cdot \frac{1}{c}=\frac{a}{bc}\]

\[-a\left(x-1\right)\ne -ax-a\]

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So, what make this an error? Well, when it comes to dealing with parenthesis, you must make sure that the product of \(2\) negatives is always going to be a positive. Here is a good example: \[\begin{matrix} -a\left(x-1\right)&=(-a)(x)+(-a)(-1)\\ &=(-ax)+(+1a)\\ &=-ax+a \end{matrix}\]

\[\left(x+a\right)^2\ne x^2+a^2\]

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So, what make this an error? Well, when it comes to dealing with parenthesis, you must make sure that you square the quantity as a whole and not as individual terms. Keep in mind that the perfect square formula is \[\left(a+b\right)^2=a^2+2ab+b^2\] so if \[\left(x+a\right)^2,\] then \[\left(x+a\right)^2=x^2+2xa+a^2\] by using the perfect square formula.

\[2\left(x+1\right)^2\ne \left(2x+2\right)^2\]

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So, what make this an error? Well, when it comes to dealing with parenthesis, you must make sure that you first square then distribute. Here is a good example: \[\begin{matrix} 2\left(x+1\right)^2&&\left(2x+2\right)^2\\ =2\left(x+1\right)\left(x+1\right)&&=\left(2x\right)^2+2(2x)(2)+2^2\\ =2(x^2+2(x)(1)+1^2)&&=\left(2x\right)\left(2x\right)+4x(2)+4\\ =2(x^2+2x+1)&&=4x^2+8x+4\\ =2x^2+4x+2&&=2(2x^2+4x+2)\\ &2x^2+4x+2\neq 4x^2+8x+4& \end{matrix}\]

\[\sqrt{x+a}\ne \sqrt{x}+\sqrt{a}\]

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So, what make this an error? Well, when it comes to dealing with square roots, you must make sure that you do not separate the internal terms. Here is a good example: \[5=\sqrt{25}=\sqrt{9+16}=\sqrt{3^2+4^2}\ne \sqrt{3^2}+\sqrt{4^2}=3+4=7\]

\[\sqrt{-x^2+a^2}\ne -\sqrt{x^2+a^2}\]

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So, what make this an error? Well, when it comes to dealing with square roots, you must make sure that you take out the negative of one term from another. Here is a good example: \[\sqrt{-x^2+a^2}=\left(-x^2+a^2\right)^{\frac{1}{2}}\]