\(\begin{cases} \frac{2 \cdot c}{\left(-b\right) - \sqrt{{b}^2 - \left(a \cdot c\right) \cdot 4}} & \text{when } b \ge 0 \\ \frac{4}{-2} \cdot \frac{\frac{b}{a}}{2} & \text{otherwise} \end{cases}\)
- Started with
\[\begin{cases} \frac{2 \cdot c}{\left(-b\right) - \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}} & \text{when } b \ge 0 \\ \frac{\left(-b\right) + \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} & \text{otherwise} \end{cases}\]
16.2
- Using strategy
rm 16.2
- Applied flip-+ to get
\[\begin{cases} \frac{2 \cdot c}{\left(-b\right) - \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}} & \text{when } b \ge 0 \\ \frac{\left(-b\right) + \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} & \text{otherwise} \end{cases} \leadsto \begin{cases} \frac{2 \cdot c}{\left(-b\right) - \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}} & \text{when } b \ge 0 \\ \frac{\frac{{\left(-b\right)}^2 - {\left(\sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}\right)}^2}{\left(-b\right) - \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}}}{2 \cdot a} & \text{otherwise} \end{cases}\]
19.8
- Applied simplify to get
\[\begin{cases} \frac{2 \cdot c}{\left(-b\right) - \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}} & \text{when } b \ge 0 \\ \frac{\frac{{\left(-b\right)}^2 - {\left(\sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}\right)}^2}{\left(-b\right) - \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}}}{2 \cdot a} & \text{otherwise} \end{cases} \leadsto \begin{cases} \frac{2 \cdot c}{\left(-b\right) - \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}} & \text{when } b \ge 0 \\ \frac{\frac{\left(4 \cdot a\right) \cdot c}{\left(-b\right) - \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}}}{2 \cdot a} & \text{otherwise} \end{cases}\]
19.8
- Applied taylor to get
\[\begin{cases} \frac{2 \cdot c}{\left(-b\right) - \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}} & \text{when } b \ge 0 \\ \frac{\frac{\left(4 \cdot a\right) \cdot c}{\left(-b\right) - \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}}}{2 \cdot a} & \text{otherwise} \end{cases} \leadsto \begin{cases} \frac{2 \cdot c}{\left(-b\right) - \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}} & \text{when } b \ge 0 \\ \frac{\frac{\left(4 \cdot a\right) \cdot c}{-2 \cdot \frac{c \cdot a}{b}}}{2 \cdot a} & \text{otherwise} \end{cases}\]
9.0
- Taylor expanded around -inf to get
\[\begin{cases} \frac{2 \cdot c}{\left(-b\right) - \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}} & \text{when } b \ge 0 \\ \frac{\frac{\left(4 \cdot a\right) \cdot c}{-2 \cdot \frac{c \cdot a}{b}}}{2 \cdot a} & \text{otherwise} \end{cases} \leadsto \begin{cases} \frac{2 \cdot c}{\left(-b\right) - \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}} & \text{when } b \ge 0 \\ \frac{\frac{\left(4 \cdot a\right) \cdot c}{-2 \cdot \frac{c \cdot a}{b}}}{2 \cdot a} & \text{otherwise} \end{cases}\]
9.0
- Applied simplify to get
\[\begin{cases} \frac{2 \cdot c}{\left(-b\right) - \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}} & \text{when } b \ge 0 \\ \frac{\frac{\left(4 \cdot a\right) \cdot c}{-2 \cdot \frac{c \cdot a}{b}}}{2 \cdot a} & \text{otherwise} \end{cases} \leadsto \begin{cases} \frac{c \cdot 2}{\left(-b\right) - \sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)}} & \text{when } b \ge 0 \\ \frac{4 \cdot a}{2 \cdot -2} \cdot \frac{\frac{1}{a}}{\frac{a}{b}} & \text{otherwise} \end{cases}\]
8.2
- Applied final simplification
- Applied simplify to get
\[\color{red}{\begin{cases} \frac{c \cdot 2}{\left(-b\right) - \sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)}} & \text{when } b \ge 0 \\ \frac{4 \cdot a}{2 \cdot -2} \cdot \frac{\frac{1}{a}}{\frac{a}{b}} & \text{otherwise} \end{cases}} \leadsto \color{blue}{\begin{cases} \frac{2 \cdot c}{\left(-b\right) - \sqrt{{b}^2 - \left(a \cdot c\right) \cdot 4}} & \text{when } b \ge 0 \\ \frac{4}{-2} \cdot \frac{\frac{b}{a}}{2} & \text{otherwise} \end{cases}}\]
3.9
