if b < -1.6322467053286328e+86 or -0.004845025334922612 < b < -1.0745984396746834e-89

1. Started with
   \[\frac{\left(-b\right) - \sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
   58.9
2. Applied taylor to get
   \[\frac{\left(-b\right) - \sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \leadsto \frac{-2 \cdot \frac{c \cdot a}{b}}{2 \cdot a}\]
   15.7
3. Taylor expanded around -inf to get
   \[\frac{\color{red}{-2 \cdot \frac{c \cdot a}{b}}}{2 \cdot a} \leadsto \frac{\color{blue}{-2 \cdot \frac{c \cdot a}{b}}}{2 \cdot a}\]
   15.7
4. Applied simplify to get
   \[\color{red}{\frac{-2 \cdot \frac{c \cdot a}{b}}{2 \cdot a}} \leadsto \color{blue}{\frac{c}{b} \cdot \frac{-2}{2}}\]
   0

if -1.6322467053286328e+86 < b < -0.004845025334922612

1. Started with
   \[\frac{\left(-b\right) - \sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
   47.6
2. Using strategy rm
   47.6
3. Applied flip-- to get
   \[\frac{\color{red}{\left(-b\right) - \sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)}}}{2 \cdot a} \leadsto \frac{\color{blue}{\frac{{\left(-b\right)}^2 - {\left(\sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)}\right)}^2}{\left(-b\right) + \sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)}}}}{2 \cdot a}\]
   47.7
4. Applied associate-/l/ to get
   \[\color{red}{\frac{\frac{{\left(-b\right)}^2 - {\left(\sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)}\right)}^2}{\left(-b\right) + \sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)}}}{2 \cdot a}} \leadsto \color{blue}{\frac{{\left(-b\right)}^2 - {\left(\sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)}\right)}^2}{\left(2 \cdot a\right) \cdot \left(\left(-b\right) + \sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)}\right)}}\]
   49.7
5. Applied taylor to get
   \[\frac{{\left(-b\right)}^2 - {\left(\sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)}\right)}^2}{\left(2 \cdot a\right) \cdot \left(\left(-b\right) + \sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)}\right)} \leadsto \frac{4 \cdot \left(c \cdot a\right)}{\left(2 \cdot a\right) \cdot \left(\left(-b\right) + \sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)}\right)}\]
   13.5
6. Taylor expanded around inf to get
   \[\frac{\color{red}{4 \cdot \left(c \cdot a\right)}}{\left(2 \cdot a\right) \cdot \left(\left(-b\right) + \sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)}\right)} \leadsto \frac{\color{blue}{4 \cdot \left(c \cdot a\right)}}{\left(2 \cdot a\right) \cdot \left(\left(-b\right) + \sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)}\right)}\]
   13.5
7. Applied simplify to get
   \[\frac{4 \cdot \left(c \cdot a\right)}{\left(2 \cdot a\right) \cdot \left(\left(-b\right) + \sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)}\right)} \leadsto \frac{\left(c \cdot 4\right) \cdot a}{(\left(\sqrt{b \cdot b - \left(c \cdot 4\right) \cdot a}\right) * \left(2 \cdot a\right) + \left(\left(-b\right) \cdot \left(2 \cdot a\right)\right))_*}\]
   13.6

---

8. Applied final simplification

if -1.0745984396746834e-89 < b < 3.741572576046744e+123

1. Started with
   \[\frac{\left(-b\right) - \sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
   11.7
2. Using strategy rm
   11.7
3. Applied div-sub to get
   \[\color{red}{\frac{\left(-b\right) - \sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}} \leadsto \color{blue}{\frac{-b}{2 \cdot a} - \frac{\sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\]
   11.7

if 3.741572576046744e+123 < b

1. Started with
   \[\frac{\left(-b\right) - \sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
   51.0
2. Using strategy rm
   51.0
3. Applied div-sub to get
   \[\color{red}{\frac{\left(-b\right) - \sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}} \leadsto \color{blue}{\frac{-b}{2 \cdot a} - \frac{\sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\]
   51.0
4. Applied taylor to get
   \[\frac{-b}{2 \cdot a} - \frac{\sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \leadsto \frac{-b}{2 \cdot a} - \frac{b - 2 \cdot \frac{c \cdot a}{b}}{2 \cdot a}\]
   12.7
5. Taylor expanded around inf to get
   \[\frac{-b}{2 \cdot a} - \frac{\color{red}{b - 2 \cdot \frac{c \cdot a}{b}}}{2 \cdot a} \leadsto \frac{-b}{2 \cdot a} - \frac{\color{blue}{b - 2 \cdot \frac{c \cdot a}{b}}}{2 \cdot a}\]
   12.7
6. Applied simplify to get
   \[\color{red}{\frac{-b}{2 \cdot a} - \frac{b - 2 \cdot \frac{c \cdot a}{b}}{2 \cdot a}} \leadsto \color{blue}{\left(\frac{\frac{c}{b}}{1} + \frac{-b}{2 \cdot a}\right) - \frac{\frac{b}{2}}{a}}\]
   0.0
7. Applied simplify to get
   \[\color{red}{\left(\frac{\frac{c}{b}}{1} + \frac{-b}{2 \cdot a}\right)} - \frac{\frac{b}{2}}{a} \leadsto \color{blue}{\left(\frac{c}{b} + \frac{-b}{2 \cdot a}\right)} - \frac{\frac{b}{2}}{a}\]
   0.0