Common math Errors

Error in mathematics

Today I will show you Common mistake made in Algebraic. and ways to go about them

Error    Correct/Justification/Example

1. $$ \frac{2}{0} \neq 0 \text{ and } \frac{2}{0} \neq 2 $$ Division by zero is undefined
2. \begin{align}  -3^2 & \neq  9 \\ \text{But} \\ (-3)^2 & = 9 \\ \text{ and} \\ -3^2 & = -9 \end{align} Watch parenthesis!
3. \begin{align} (x^2)^3 & \neq  x^5 \quad \text{but} \\ (x^2)^3 & = x^2x^2x^2 \\ & = x^6 \end{align}
4. \begin{align} \frac{a}{b+c} \neq & \frac{a}{b} + \frac{a}{c} \\ _ \text{Clear Example below} \\ \rightarrow\ \frac{1}{2} & = \frac{1}{1+1} \\ \text{but} \quad \frac{1}{1+1} & \neq \frac{1}{1} + \frac{1}{1} \\ \frac{1}{1} + \frac{1}{1} & = 2 \end{align}
5. $$ \frac{1}{x^2+x^3} \neq x^{-2} + x^{-3} $$ A more complex of the previous error
6. $$ -a(x-1) \neq -ax-a \quad \text{but} \\ {-a(x-1) = -ax+a } $$ Make sure you distribute the "-"!
7. \begin{align} (x+a)^2 & \neq x^2+a^2 \quad \text{but} \\ (x+a)^2 & = (x+a)(x+a) \\  & = x^2+2xa+a2 \end{align}
8. \begin{align} \require{cancel} \frac{\cancel{a}+bx}{\cancel{a}} & \neq -ax-b \quad \text{but} \\ \frac{a+bx}{a} & = \frac{a}{a}+\frac{bx}{a} \\  & = 1 + \frac{bx}{a} \end{align} Beware of incorrect canceling
9. \begin{align} \sqrt{x^2 + a^2} & \neq  x + a \\ _ \text{Proof} \\ 5 & = \sqrt{25} \\ \sqrt{3^2 + 4^2} & = \sqrt{3^2} + \sqrt{4^2} \\ & = 3+4 \\ & = 7 \\ \text{and } 5 & \neq 7 \end{align}
10. $$ \sqrt{x+a} \neq \sqrt{x}+ \sqrt{a} $$ See previous error
11. $$ (x+a)^n \neq x^n+a^n  \text{   and   } \\ {\sqrt[n]{x+a} \neq \sqrt[n]{x} + \sqrt[n]{a}} $$ More general versions of previous three errors
12. \begin{align} 2(x+1)^2  & \neq   (2x+2)^2 \\ _ \text{Proof} \\ 2(x+1)^2 & = 2(x^2+2x+1) \\ & = 2x^2+4x+2 \end{align} Square first then distribute!
13. $$ (2x+2)^2 \neq 2(x+1)^2 $$ See the previous example. you can not factor out a constant if there is a power on the parenthesis!
14. \begin{align} \sqrt{-x^2 + a^2} & \neq -\sqrt{x^2 + a^2} \\ _ \text{But} \\ \sqrt{-x^2 + a^2} & = (-x^2 + a^2)^{\frac{1}{2}} \end{align} Now see the previous error
15. \begin{align} \frac{a}{\left (\frac{b}{c} \right)} & \neq \frac{ab}{c} \\ _ \text{but} \\ \frac{a}{ \left (\frac{b}{c} \right) } & =\left(\frac{a}{b} \right)\left(\frac{1}{c} \right) \\ & = \frac{ac}{b} \end{align}
16. \begin{align} \frac{\left (\frac{a}{b} \right)}{c} & \neq \frac{ac}{b} \\ _ \text{but} \\ \frac{\left (\frac{a}{b} \right)}{c} & =\left(\frac{a}{b} \right)\left(\frac{1}{c} \right) \\ & = \frac{a}{bc} \end{align}

Question misinterpret based on the way they look

Mostly question posted on the Internet is been misunderstood.

1.  Imaging someone post this \( 3\sqrt{x}\) as 3rootx

3rootx can be  \( 3\sqrt{x}\) or \( \sqrt[3]{x}\)

 $$ 3\sqrt{x} = 9^{\frac{1}{2}} x^{\frac{1}{2}} = \left(9x \right)^{\frac{1}{2}} = \sqrt{9x} $$

and $$ \sqrt[3]{x} = x^{ \frac{1}{3} } $$
2.  Real question \( \frac{a+2}{a-2} \) posted as a+2/a-2
to me this a+2/a-2 mean \( a + \frac{2}{a} - 2\) using parentheses ( ) will give make it easy to understand. Eg ( a+2 )/( a-2 )

Nb:

1.  Order of Operation

• Do things in parentheses first
• Exponents
• Multiplication and Division
• Addition and Subtraction

2.   Master the Key Concepts

• Do not try to memorize the processes
• Understand the process and logic that is involved