\(\begin{cases} \frac{\left(-b\right) - \sqrt{{b}^2 - c \cdot \left(4 \cdot a\right)}}{a \cdot 2} & \text{when } b \ge 0 \\ \frac{\frac{-2}{a} \cdot \frac{c}{4}}{\frac{b}{a \cdot 2}} & \text{otherwise} \end{cases}\)
- Started with
\[\begin{cases} \frac{\left(-b\right) - \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} & \text{when } b \ge 0 \\ \frac{2 \cdot c}{\left(-b\right) + \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}} & \text{otherwise} \end{cases}\]
30.5
- Using strategy
rm 30.5
- Applied flip-+ to get
\[\begin{cases} \frac{\left(-b\right) - \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} & \text{when } b \ge 0 \\ \color{red}{\frac{2 \cdot c}{\left(-b\right) + \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}}} & \text{otherwise} \end{cases} \leadsto \begin{cases} \frac{\left(-b\right) - \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} & \text{when } b \ge 0 \\ \color{blue}{\frac{2 \cdot c}{\frac{{\left(-b\right)}^2 - {\left(\sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}\right)}^2}{\left(-b\right) - \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}}}} & \text{otherwise} \end{cases}\]
44.9
- Applied simplify to get
\[\begin{cases} \frac{\left(-b\right) - \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} & \text{when } b \ge 0 \\ \frac{\color{red}{2 \cdot c}}{\frac{{\left(-b\right)}^2 - {\left(\sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}\right)}^2}{\left(-b\right) - \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}}} & \text{otherwise} \end{cases} \leadsto \begin{cases} \frac{\left(-b\right) - \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} & \text{when } b \ge 0 \\ \frac{\color{blue}{2 \cdot c}}{\frac{\left(4 \cdot a\right) \cdot c}{\left(-b\right) - \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}}} & \text{otherwise} \end{cases}\]
44.1
- Applied taylor to get
\[\begin{cases} \frac{\left(-b\right) - \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} & \text{when } b \ge 0 \\ \frac{2 \cdot c}{\frac{\left(4 \cdot a\right) \cdot c}{\left(-b\right) - \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}}} & \text{otherwise} \end{cases} \leadsto \begin{cases} \frac{\left(-b\right) - \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} & \text{when } b \ge 0 \\ \frac{2 \cdot c}{\frac{\left(4 \cdot a\right) \cdot c}{-2 \cdot \frac{c \cdot a}{b}}} & \text{otherwise} \end{cases}\]
20.7
- Taylor expanded around -inf to get
\[\begin{cases} \frac{\left(-b\right) - \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} & \text{when } b \ge 0 \\ \frac{2 \cdot c}{\color{red}{\frac{\left(4 \cdot a\right) \cdot c}{-2 \cdot \frac{c \cdot a}{b}}}} & \text{otherwise} \end{cases} \leadsto \begin{cases} \frac{\left(-b\right) - \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} & \text{when } b \ge 0 \\ \frac{2 \cdot c}{\color{blue}{\frac{\left(4 \cdot a\right) \cdot c}{-2 \cdot \frac{c \cdot a}{b}}}} & \text{otherwise} \end{cases}\]
20.7
- Applied simplify to get
\[\begin{cases} \frac{\left(-b\right) - \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} & \text{when } b \ge 0 \\ \frac{2 \cdot c}{\frac{\left(4 \cdot a\right) \cdot c}{-2 \cdot \frac{c \cdot a}{b}}} & \text{otherwise} \end{cases} \leadsto \begin{cases} \frac{\left(-b\right) - \sqrt{{b}^2 - a \cdot \left(c \cdot 4\right)}}{2 \cdot a} & \text{when } b \ge 0 \\ \left(\frac{c}{4} \cdot \frac{-2}{a}\right) \cdot \frac{\frac{2}{1}}{\frac{b}{a}} & \text{otherwise} \end{cases}\]
18.9
- Applied final simplification
- Applied simplify to get
\[\color{red}{\begin{cases} \frac{\left(-b\right) - \sqrt{{b}^2 - a \cdot \left(c \cdot 4\right)}}{2 \cdot a} & \text{when } b \ge 0 \\ \left(\frac{c}{4} \cdot \frac{-2}{a}\right) \cdot \frac{\frac{2}{1}}{\frac{b}{a}} & \text{otherwise} \end{cases}} \leadsto \color{blue}{\begin{cases} \frac{\left(-b\right) - \sqrt{{b}^2 - c \cdot \left(4 \cdot a\right)}}{a \cdot 2} & \text{when } b \ge 0 \\ \frac{\frac{-2}{a} \cdot \frac{c}{4}}{\frac{b}{a \cdot 2}} & \text{otherwise} \end{cases}}\]
18.8
