if b < -4.191830935951668e+150 or 3.4791896352684183e+99 < b

1. Started with
   \[\begin{cases} \frac{\left(-b\right) - \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} & \text{when } b \ge 0 \\ \frac{2 \cdot c}{\left(-b\right) + \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}} & \text{otherwise} \end{cases}\]
   41.8
2. Using strategy rm
   41.8
3. Applied add-sqr-sqrt to get
   \[\begin{cases} \frac{\left(-b\right) - \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} & \text{when } b \ge 0 \\ \frac{2 \cdot c}{\color{red}{\left(-b\right) + \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}}} & \text{otherwise} \end{cases} \leadsto \begin{cases} \frac{\left(-b\right) - \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} & \text{when } b \ge 0 \\ \frac{2 \cdot c}{\color{blue}{\left(-b\right) + {\left(\sqrt{\sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}}\right)}^2}} & \text{otherwise} \end{cases}\]
   41.8
4. Applied taylor to get
   \[\begin{cases} \frac{\left(-b\right) - \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} & \text{when } b \ge 0 \\ \frac{2 \cdot c}{\left(-b\right) + {\left(\sqrt{\sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}}\right)}^2} & \text{otherwise} \end{cases} \leadsto \begin{cases} \frac{\left(-b\right) - \left(b - 2 \cdot \frac{c \cdot a}{b}\right)}{2 \cdot a} & \text{when } b \ge 0 \\ \frac{2 \cdot c}{\left(-b\right) + {\left(\sqrt{\sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}}\right)}^2} & \text{otherwise} \end{cases}\]
   25.8
5. Taylor expanded around inf to get
   \[\begin{cases} \frac{\left(-b\right) - \color{red}{\left(b - 2 \cdot \frac{c \cdot a}{b}\right)}}{2 \cdot a} & \text{when } b \ge 0 \\ \frac{2 \cdot c}{\left(-b\right) + {\left(\sqrt{\sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}}\right)}^2} & \text{otherwise} \end{cases} \leadsto \begin{cases} \frac{\left(-b\right) - \color{blue}{\left(b - 2 \cdot \frac{c \cdot a}{b}\right)}}{2 \cdot a} & \text{when } b \ge 0 \\ \frac{2 \cdot c}{\left(-b\right) + {\left(\sqrt{\sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}}\right)}^2} & \text{otherwise} \end{cases}\]
   25.8
6. Applied simplify to get
   \[\color{red}{\begin{cases} \frac{\left(-b\right) - \left(b - 2 \cdot \frac{c \cdot a}{b}\right)}{2 \cdot a} & \text{when } b \ge 0 \\ \frac{2 \cdot c}{\left(-b\right) + {\left(\sqrt{\sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}}\right)}^2} & \text{otherwise} \end{cases}} \leadsto \color{blue}{\begin{cases} \frac{\left(\left(-b\right) - b\right) + \left(2 \cdot a\right) \cdot \frac{c}{b}}{2 \cdot a} & \text{when } b \ge 0 \\ \frac{2 \cdot c}{\sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)} + \left(-b\right)} & \text{otherwise} \end{cases}}\]
   21.5
7. Applied taylor to get
   \[\begin{cases} \frac{\left(\left(-b\right) - b\right) + \left(2 \cdot a\right) \cdot \frac{c}{b}}{2 \cdot a} & \text{when } b \ge 0 \\ \frac{2 \cdot c}{\sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)} + \left(-b\right)} & \text{otherwise} \end{cases} \leadsto \begin{cases} \frac{\left(\left(-b\right) - b\right) + \left(2 \cdot a\right) \cdot \frac{c}{b}}{2 \cdot a} & \text{when } b \ge 0 \\ \frac{2 \cdot c}{\left(2 \cdot \frac{c \cdot a}{b} - b\right) + \left(-b\right)} & \text{otherwise} \end{cases}\]
   4.8
8. Taylor expanded around -inf to get
   \[\begin{cases} \frac{\left(\left(-b\right) - b\right) + \left(2 \cdot a\right) \cdot \frac{c}{b}}{2 \cdot a} & \text{when } b \ge 0 \\ \frac{\color{red}{2 \cdot c}}{\left(2 \cdot \frac{c \cdot a}{b} - b\right) + \left(-b\right)} & \text{otherwise} \end{cases} \leadsto \begin{cases} \frac{\left(\left(-b\right) - b\right) + \left(2 \cdot a\right) \cdot \frac{c}{b}}{2 \cdot a} & \text{when } b \ge 0 \\ \frac{\color{blue}{2 \cdot c}}{\left(2 \cdot \frac{c \cdot a}{b} - b\right) + \left(-b\right)} & \text{otherwise} \end{cases}\]
   4.8
9. Applied simplify to get
   \[\begin{cases} \frac{\left(\left(-b\right) - b\right) + \left(2 \cdot a\right) \cdot \frac{c}{b}}{2 \cdot a} & \text{when } b \ge 0 \\ \frac{2 \cdot c}{\left(2 \cdot \frac{c \cdot a}{b} - b\right) + \left(-b\right)} & \text{otherwise} \end{cases} \leadsto \begin{cases} \frac{\left(a \cdot 2\right) \cdot \frac{c}{b} - \left(b - \left(-b\right)\right)}{a \cdot 2} & \text{when } b \ge 0 \\ \frac{c \cdot 2}{\left(a \cdot 2\right) \cdot \frac{c}{b} - \left(b - \left(-b\right)\right)} & \text{otherwise} \end{cases}\]
   1.5

---

10. Applied final simplification

if -4.191830935951668e+150 < b < 4.2941756400879677e-302

1. Started with
   \[\begin{cases} \frac{\left(-b\right) - \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} & \text{when } b \ge 0 \\ \frac{2 \cdot c}{\left(-b\right) + \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}} & \text{otherwise} \end{cases}\]
   9.7
2. Using strategy rm
   9.7
3. Applied flip-- to get
   \[\begin{cases} \frac{\color{red}{\left(-b\right) - \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}}}{2 \cdot a} & \text{when } b \ge 0 \\ \frac{2 \cdot c}{\left(-b\right) + \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}} & \text{otherwise} \end{cases} \leadsto \begin{cases} \frac{\color{blue}{\frac{{\left(-b\right)}^2 - {\left(\sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}\right)}^2}{\left(-b\right) + \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}}}}{2 \cdot a} & \text{when } b \ge 0 \\ \frac{2 \cdot c}{\left(-b\right) + \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}} & \text{otherwise} \end{cases}\]
   9.7
4. Applied simplify to get
   \[\begin{cases} \frac{\frac{\color{red}{{\left(-b\right)}^2 - {\left(\sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}\right)}^2}}{\left(-b\right) + \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}}}{2 \cdot a} & \text{when } b \ge 0 \\ \frac{2 \cdot c}{\left(-b\right) + \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}} & \text{otherwise} \end{cases} \leadsto \begin{cases} \frac{\frac{\color{blue}{\left(4 \cdot a\right) \cdot c}}{\left(-b\right) + \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}}}{2 \cdot a} & \text{when } b \ge 0 \\ \frac{2 \cdot c}{\left(-b\right) + \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}} & \text{otherwise} \end{cases}\]
   9.7
5. Applied taylor to get
   \[\begin{cases} \frac{\frac{\left(4 \cdot a\right) \cdot c}{\left(-b\right) + \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}}}{2 \cdot a} & \text{when } b \ge 0 \\ \frac{2 \cdot c}{\left(-b\right) + \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}} & \text{otherwise} \end{cases} \leadsto \begin{cases} \frac{-2 \cdot b}{2 \cdot a} & \text{when } b \ge 0 \\ \frac{2 \cdot c}{\left(-b\right) + \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}} & \text{otherwise} \end{cases}\]
   8.7
6. Taylor expanded around 0 to get
   \[\begin{cases} \frac{\color{red}{-2 \cdot b}}{2 \cdot a} & \text{when } b \ge 0 \\ \frac{2 \cdot c}{\left(-b\right) + \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}} & \text{otherwise} \end{cases} \leadsto \begin{cases} \frac{\color{blue}{-2 \cdot b}}{2 \cdot a} & \text{when } b \ge 0 \\ \frac{2 \cdot c}{\left(-b\right) + \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}} & \text{otherwise} \end{cases}\]
   8.7
7. Applied simplify to get
   \[\begin{cases} \frac{-2 \cdot b}{2 \cdot a} & \text{when } b \ge 0 \\ \frac{2 \cdot c}{\left(-b\right) + \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}} & \text{otherwise} \end{cases} \leadsto \begin{cases} \frac{b}{a} \cdot \frac{-2}{2} & \text{when } b \ge 0 \\ \frac{2 \cdot c}{\left(-b\right) + \sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)}} & \text{otherwise} \end{cases}\]
   8.6

---

8. Applied final simplification

if 4.2941756400879677e-302 < b < 3.4791896352684183e+99

1. Started with
   \[\begin{cases} \frac{\left(-b\right) - \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} & \text{when } b \ge 0 \\ \frac{2 \cdot c}{\left(-b\right) + \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}} & \text{otherwise} \end{cases}\]
   8.8
2. Using strategy rm
   8.8
3. Applied flip-+ to get
   \[\begin{cases} \frac{\left(-b\right) - \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} & \text{when } b \ge 0 \\ \color{red}{\frac{2 \cdot c}{\left(-b\right) + \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}}} & \text{otherwise} \end{cases} \leadsto \begin{cases} \frac{\left(-b\right) - \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} & \text{when } b \ge 0 \\ \color{blue}{\frac{2 \cdot c}{\frac{{\left(-b\right)}^2 - {\left(\sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}\right)}^2}{\left(-b\right) - \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}}}} & \text{otherwise} \end{cases}\]
   8.8
4. Applied simplify to get
   \[\begin{cases} \frac{\left(-b\right) - \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} & \text{when } b \ge 0 \\ \frac{\color{red}{2 \cdot c}}{\frac{{\left(-b\right)}^2 - {\left(\sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}\right)}^2}{\left(-b\right) - \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}}} & \text{otherwise} \end{cases} \leadsto \begin{cases} \frac{\left(-b\right) - \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} & \text{when } b \ge 0 \\ \frac{\color{blue}{2 \cdot c}}{\frac{\left(4 \cdot a\right) \cdot c}{\left(-b\right) - \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}}} & \text{otherwise} \end{cases}\]
   8.8
5. Applied taylor to get
   \[\begin{cases} \frac{\left(-b\right) - \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} & \text{when } b \ge 0 \\ \frac{2 \cdot c}{\frac{\left(4 \cdot a\right) \cdot c}{\left(-b\right) - \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}}} & \text{otherwise} \end{cases} \leadsto \begin{cases} \frac{\left(-b\right) - \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} & \text{when } b \ge 0 \\ \frac{2 \cdot c}{\frac{\left(4 \cdot a\right) \cdot c}{-2 \cdot \frac{c \cdot a}{b}}} & \text{otherwise} \end{cases}\]
   8.8
6. Taylor expanded around -inf to get
   \[\begin{cases} \frac{\left(-b\right) - \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} & \text{when } b \ge 0 \\ \frac{2 \cdot c}{\color{red}{\frac{\left(4 \cdot a\right) \cdot c}{-2 \cdot \frac{c \cdot a}{b}}}} & \text{otherwise} \end{cases} \leadsto \begin{cases} \frac{\left(-b\right) - \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} & \text{when } b \ge 0 \\ \frac{2 \cdot c}{\color{blue}{\frac{\left(4 \cdot a\right) \cdot c}{-2 \cdot \frac{c \cdot a}{b}}}} & \text{otherwise} \end{cases}\]
   8.8
7. Applied simplify to get
   \[\begin{cases} \frac{\left(-b\right) - \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} & \text{when } b \ge 0 \\ \frac{2 \cdot c}{\frac{\left(4 \cdot a\right) \cdot c}{-2 \cdot \frac{c \cdot a}{b}}} & \text{otherwise} \end{cases} \leadsto \begin{cases} \frac{\left(-b\right) - \sqrt{{b}^2 - a \cdot \left(c \cdot 4\right)}}{2 \cdot a} & \text{when } b \ge 0 \\ \left(\frac{c}{4} \cdot \frac{-2}{a}\right) \cdot \frac{\frac{2}{1}}{\frac{b}{a}} & \text{otherwise} \end{cases}\]
   8.7

---

8. Applied final simplification
9. Applied simplify to get
   \[\color{red}{\begin{cases} \frac{\left(-b\right) - \sqrt{{b}^2 - a \cdot \left(c \cdot 4\right)}}{2 \cdot a} & \text{when } b \ge 0 \\ \left(\frac{c}{4} \cdot \frac{-2}{a}\right) \cdot \frac{\frac{2}{1}}{\frac{b}{a}} & \text{otherwise} \end{cases}} \leadsto \color{blue}{\begin{cases} \frac{\left(-b\right) - \sqrt{{b}^2 - c \cdot \left(4 \cdot a\right)}}{a \cdot 2} & \text{when } b \ge 0 \\ \frac{\frac{-2}{a} \cdot \frac{c}{4}}{\frac{b}{a \cdot 2}} & \text{otherwise} \end{cases}}\]
   8.8