if b < -8.642142288968947e+88

1. Started with
   \[\frac{\left(-b\right) - \sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
   58.9
2. Using strategy rm
   58.9
3. Applied flip-- to get
   \[\frac{\color{red}{\left(-b\right) - \sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)}}}{2 \cdot a} \leadsto \frac{\color{blue}{\frac{{\left(-b\right)}^2 - {\left(\sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)}\right)}^2}{\left(-b\right) + \sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)}}}}{2 \cdot a}\]
   59.0
4. Applied simplify to get
   \[\frac{\frac{\color{red}{{\left(-b\right)}^2 - {\left(\sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)}\right)}^2}}{\left(-b\right) + \sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)}}}{2 \cdot a} \leadsto \frac{\frac{\color{blue}{c \cdot \left(4 \cdot a\right)}}{\left(-b\right) + \sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)}}}{2 \cdot a}\]
   34.8
5. Applied taylor to get
   \[\frac{\frac{c \cdot \left(4 \cdot a\right)}{\left(-b\right) + \sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)}}}{2 \cdot a} \leadsto \frac{\frac{c \cdot \left(4 \cdot a\right)}{\left(-b\right) + \left(2 \cdot \frac{c \cdot a}{b} - b\right)}}{2 \cdot a}\]
   15.9
6. Taylor expanded around -inf to get
   \[\frac{\frac{c \cdot \left(4 \cdot a\right)}{\left(-b\right) + \color{red}{\left(2 \cdot \frac{c \cdot a}{b} - b\right)}}}{2 \cdot a} \leadsto \frac{\frac{c \cdot \left(4 \cdot a\right)}{\left(-b\right) + \color{blue}{\left(2 \cdot \frac{c \cdot a}{b} - b\right)}}}{2 \cdot a}\]
   15.9
7. Applied simplify to get
   \[\color{red}{\frac{\frac{c \cdot \left(4 \cdot a\right)}{\left(-b\right) + \left(2 \cdot \frac{c \cdot a}{b} - b\right)}}{2 \cdot a}} \leadsto \color{blue}{\frac{\frac{4 \cdot c}{\frac{2 \cdot a}{\frac{b}{c}} - \left(b - \left(-b\right)\right)}}{2}}\]
   1.1

if -8.642142288968947e+88 < b < -3.247958591749237e-290

1. Started with
   \[\frac{\left(-b\right) - \sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
   32.7
2. Using strategy rm
   32.7
3. Applied flip-- to get
   \[\frac{\color{red}{\left(-b\right) - \sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)}}}{2 \cdot a} \leadsto \frac{\color{blue}{\frac{{\left(-b\right)}^2 - {\left(\sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)}\right)}^2}{\left(-b\right) + \sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)}}}}{2 \cdot a}\]
   32.8
4. Applied simplify to get
   \[\frac{\frac{\color{red}{{\left(-b\right)}^2 - {\left(\sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)}\right)}^2}}{\left(-b\right) + \sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)}}}{2 \cdot a} \leadsto \frac{\frac{\color{blue}{c \cdot \left(4 \cdot a\right)}}{\left(-b\right) + \sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)}}}{2 \cdot a}\]
   17.0
5. Using strategy rm
   17.0
6. Applied clear-num to get
   \[\color{red}{\frac{\frac{c \cdot \left(4 \cdot a\right)}{\left(-b\right) + \sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)}}}{2 \cdot a}} \leadsto \color{blue}{\frac{1}{\frac{2 \cdot a}{\frac{c \cdot \left(4 \cdot a\right)}{\left(-b\right) + \sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)}}}}}\]
   17.1
7. Applied simplify to get
   \[\frac{1}{\color{red}{\frac{2 \cdot a}{\frac{c \cdot \left(4 \cdot a\right)}{\left(-b\right) + \sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)}}}}} \leadsto \frac{1}{\color{blue}{\left(\left(-b\right) + \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}\right) \cdot \left(\frac{1}{c} \cdot \frac{2}{4}\right)}}\]
   9.1
8. Applied simplify to get
   \[\frac{1}{\color{red}{\left(\left(-b\right) + \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}\right)} \cdot \left(\frac{1}{c} \cdot \frac{2}{4}\right)} \leadsto \frac{1}{\color{blue}{\left(\left(-b\right) + \sqrt{{b}^2 - \left(a \cdot c\right) \cdot 4}\right)} \cdot \left(\frac{1}{c} \cdot \frac{2}{4}\right)}\]
   9.1
9. Applied simplify to get
   \[\frac{1}{\left(\left(-b\right) + \sqrt{{b}^2 - \left(a \cdot c\right) \cdot 4}\right) \cdot \color{red}{\left(\frac{1}{c} \cdot \frac{2}{4}\right)}} \leadsto \frac{1}{\left(\left(-b\right) + \sqrt{{b}^2 - \left(a \cdot c\right) \cdot 4}\right) \cdot \color{blue}{\frac{\frac{2}{c}}{4}}}\]
   9.1

if -3.247958591749237e-290 < b < 2.2736054037680634e+124

1. Started with
   \[\frac{\left(-b\right) - \sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
   9.2
2. Using strategy rm
   9.2
3. Applied associate-*r* to get
   \[\frac{\left(-b\right) - \sqrt{{b}^2 - \color{red}{4 \cdot \left(a \cdot c\right)}}}{2 \cdot a} \leadsto \frac{\left(-b\right) - \sqrt{{b}^2 - \color{blue}{\left(4 \cdot a\right) \cdot c}}}{2 \cdot a}\]
   9.2

if 2.2736054037680634e+124 < b

1. Started with
   \[\frac{\left(-b\right) - \sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
   51.8
2. Using strategy rm
   51.8
3. Applied add-cbrt-cube to get
   \[\frac{\left(-b\right) - \sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)}}{\color{red}{2 \cdot a}} \leadsto \frac{\left(-b\right) - \sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)}}{\color{blue}{\sqrt[3]{{\left(2 \cdot a\right)}^3}}}\]
   56.6
4. Applied add-cbrt-cube to get
   \[\frac{\color{red}{\left(-b\right) - \sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)}}}{\sqrt[3]{{\left(2 \cdot a\right)}^3}} \leadsto \frac{\color{blue}{\sqrt[3]{{\left(\left(-b\right) - \sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)}\right)}^3}}}{\sqrt[3]{{\left(2 \cdot a\right)}^3}}\]
   61.3
5. Applied cbrt-undiv to get
   \[\color{red}{\frac{\sqrt[3]{{\left(\left(-b\right) - \sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)}\right)}^3}}{\sqrt[3]{{\left(2 \cdot a\right)}^3}}} \leadsto \color{blue}{\sqrt[3]{\frac{{\left(\left(-b\right) - \sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)}\right)}^3}{{\left(2 \cdot a\right)}^3}}}\]
   61.3
6. Applied simplify to get
   \[\sqrt[3]{\color{red}{\frac{{\left(\left(-b\right) - \sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)}\right)}^3}{{\left(2 \cdot a\right)}^3}}} \leadsto \sqrt[3]{\color{blue}{{\left(\frac{\left(-b\right) - \sqrt{{b}^2 - \left(a \cdot c\right) \cdot 4}}{2 \cdot a}\right)}^3}}\]
   57.8
7. Applied taylor to get
   \[\sqrt[3]{{\left(\frac{\left(-b\right) - \sqrt{{b}^2 - \left(a \cdot c\right) \cdot 4}}{2 \cdot a}\right)}^3} \leadsto \sqrt[3]{{\left(\frac{\left(-b\right) - \left(b - 2 \cdot \frac{c \cdot a}{b}\right)}{2 \cdot a}\right)}^3}\]
   43.1
8. Taylor expanded around inf to get
   \[\sqrt[3]{{\left(\frac{\left(-b\right) - \color{red}{\left(b - 2 \cdot \frac{c \cdot a}{b}\right)}}{2 \cdot a}\right)}^3} \leadsto \sqrt[3]{{\left(\frac{\left(-b\right) - \color{blue}{\left(b - 2 \cdot \frac{c \cdot a}{b}\right)}}{2 \cdot a}\right)}^3}\]
   43.1
9. Applied simplify to get
   \[\sqrt[3]{{\left(\frac{\left(-b\right) - \left(b - 2 \cdot \frac{c \cdot a}{b}\right)}{2 \cdot a}\right)}^3} \leadsto \frac{\left(\left(-b\right) - b\right) + \frac{2 \cdot c}{\frac{b}{a}}}{a \cdot 2}\]
   1.8

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10. Applied final simplification