\(\begin{cases} \frac{\frac{4}{2} \cdot b}{-2 \cdot a} & \text{when } b \ge 0 \\ \frac{2 \cdot c}{\left(-b\right) + \sqrt{{b}^2 - \left(a \cdot c\right) \cdot 4}} & \text{otherwise} \end{cases}\)
- Started with
\[\begin{cases} \frac{\left(-b\right) - \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} & \text{when } b \ge 0 \\ \frac{2 \cdot c}{\left(-b\right) + \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}} & \text{otherwise} \end{cases}\]
16.1
- Using strategy
rm 16.1
- Applied flip-- to get
\[\begin{cases} \frac{\color{red}{\left(-b\right) - \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}}}{2 \cdot a} & \text{when } b \ge 0 \\ \frac{2 \cdot c}{\left(-b\right) + \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}} & \text{otherwise} \end{cases} \leadsto \begin{cases} \frac{\color{blue}{\frac{{\left(-b\right)}^2 - {\left(\sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}\right)}^2}{\left(-b\right) + \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}}}}{2 \cdot a} & \text{when } b \ge 0 \\ \frac{2 \cdot c}{\left(-b\right) + \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}} & \text{otherwise} \end{cases}\]
19.9
- Applied simplify to get
\[\begin{cases} \frac{\frac{\color{red}{{\left(-b\right)}^2 - {\left(\sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}\right)}^2}}{\left(-b\right) + \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}}}{2 \cdot a} & \text{when } b \ge 0 \\ \frac{2 \cdot c}{\left(-b\right) + \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}} & \text{otherwise} \end{cases} \leadsto \begin{cases} \frac{\frac{\color{blue}{\left(4 \cdot a\right) \cdot c}}{\left(-b\right) + \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}}}{2 \cdot a} & \text{when } b \ge 0 \\ \frac{2 \cdot c}{\left(-b\right) + \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}} & \text{otherwise} \end{cases}\]
19.9
- Applied taylor to get
\[\begin{cases} \frac{\frac{\left(4 \cdot a\right) \cdot c}{\left(-b\right) + \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}}}{2 \cdot a} & \text{when } b \ge 0 \\ \frac{2 \cdot c}{\left(-b\right) + \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}} & \text{otherwise} \end{cases} \leadsto \begin{cases} \frac{\frac{\left(4 \cdot a\right) \cdot c}{-2 \cdot \frac{c \cdot a}{b}}}{2 \cdot a} & \text{when } b \ge 0 \\ \frac{2 \cdot c}{\left(-b\right) + \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}} & \text{otherwise} \end{cases}\]
9.0
- Taylor expanded around inf to get
\[\begin{cases} \frac{\frac{\left(4 \cdot a\right) \cdot c}{\color{red}{-2 \cdot \frac{c \cdot a}{b}}}}{2 \cdot a} & \text{when } b \ge 0 \\ \frac{2 \cdot c}{\left(-b\right) + \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}} & \text{otherwise} \end{cases} \leadsto \begin{cases} \frac{\frac{\left(4 \cdot a\right) \cdot c}{\color{blue}{-2 \cdot \frac{c \cdot a}{b}}}}{2 \cdot a} & \text{when } b \ge 0 \\ \frac{2 \cdot c}{\left(-b\right) + \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}} & \text{otherwise} \end{cases}\]
9.0
- Applied simplify to get
\[\begin{cases} \frac{\frac{\left(4 \cdot a\right) \cdot c}{-2 \cdot \frac{c \cdot a}{b}}}{2 \cdot a} & \text{when } b \ge 0 \\ \frac{2 \cdot c}{\left(-b\right) + \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}} & \text{otherwise} \end{cases} \leadsto \begin{cases} \frac{a \cdot 4}{-2 \cdot a} \cdot \frac{\frac{1}{a}}{\frac{2}{b}} & \text{when } b \ge 0 \\ \frac{2 \cdot c}{\sqrt{{b}^2 - \left(c \cdot 4\right) \cdot a} + \left(-b\right)} & \text{otherwise} \end{cases}\]
4.2
- Applied final simplification
- Applied simplify to get
\[\color{red}{\begin{cases} \frac{a \cdot 4}{-2 \cdot a} \cdot \frac{\frac{1}{a}}{\frac{2}{b}} & \text{when } b \ge 0 \\ \frac{2 \cdot c}{\sqrt{{b}^2 - \left(c \cdot 4\right) \cdot a} + \left(-b\right)} & \text{otherwise} \end{cases}} \leadsto \color{blue}{\begin{cases} \frac{\frac{4}{2} \cdot b}{-2 \cdot a} & \text{when } b \ge 0 \\ \frac{2 \cdot c}{\left(-b\right) + \sqrt{{b}^2 - \left(a \cdot c\right) \cdot 4}} & \text{otherwise} \end{cases}}\]
3.9
