Section 1.1 Algebra
Subsection 1.1.1 Things to know
- If \(ax^2+bx+c=0\text{,}\) the roots are \(x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}\text{;}\) the sum of the roots is \(-b/a\text{;}\) the product of the roots is \(c/a\text{.}\)
- \(\displaystyle x^{-r} = \frac{1}{x^r}.\)
- \(x^r x^s=x^{r+s}\text{.}\)
- \(x^{1/n} = \sqrt[n]{x},\) for \(n\) integer. If \(n\) is even, then \(x\) must be non-negative and \(\sqrt[n]{x}\) denotes the non-negative root of \(x\text{.}\)
- \(\left( x^r \right)^s = x^{r \cdot s}\text{,}\) assuming that \(x \geq 0\text{.}\) Note that care must be taken if \(x\) is negative. For instance, \((x^2)^{1/2} \neq x\) if \(x \lt 0\text{;}\) rather, \((x^2)^{1/2} = |x|\text{.}\)
- Finally, avoid common mistakes: remember that, in general,\begin{gather*} \sqrt{x+y} \neq \sqrt{x} + \sqrt{y},\\ \frac{1}{x+y} \neq \frac{1}{x} + \frac{1}{y},\\ (x+y)^2 \neq x^2 + y^2. \end{gather*}
Subsection 1.1.2 Review videos
The following review videos may be useful: