Formulae to Solve of Polynomial Equations

The Linear Formula

$$x=\frac{-<!--\; -\; -->b}{a}$$

The linear formula gives the solution of $ax+b=0$ for real numbers $a$, $b$ with $a\ne <!--\; \ne \; -->0$.

The Quadratic Formula

$$x=\frac{-<!--\; -\; -->b\pm <!--\; \pm \; -->\sqrt{b^2-<!--\; -\; -->4ac}}{2a}$$

The quadratic formula gives the solutions of $a{x}^2+bx+c=0$ for real numbers $a$, $b$, $c$ with $a\ne <!--\; \ne \; -->0$.

The Cubic Formula

$$x=\frac{-<!--\; -\; -->2b+(\frac{-<!--\; -\; -->1+\sqrt{-<!--\; -\; -->3}}{2}{)}^n\sqrt[3]{4(-<!--\; -\; -->2b^3+9abc-<!--\; -\; -->27a^{2}d+\sqrt{(-<!--\; -\; -->2b^3+9abc-<!--\; -\; -->27a^{2}d{)}^2-<!--\; -\; -->4(b^2-<!--\; -\; -->3ac{)}^3})}+(\frac{-<!--\; -\; -->1-<!--\; -\; -->\sqrt{-<!--\; -\; -->3}}{2}{)}^n\sqrt[3]{4(-<!--\; -\; -->2b^3+9abc-<!--\; -\; -->27a^{2}d-<!--\; -\; -->\sqrt{(-<!--\; -\; -->2b^3+9abc-<!--\; -\; -->27a^{2}d{)}^2-<!--\; -\; -->4(b^2-<!--\; -\; -->3ac{)}^3})}}{6a}$$

The cubic formula gives the solutions of $a{x}^3+b{x}^2+cx+d=0$ for real numbers $a$, $b$, $c$, $d$ with $a\ne <!--\; \ne \; -->0$.

Directions: Take $n=0$, $1$, $2$. Use real cube roots if possible, and principal roots otherwise.

The Quartic Formula

$$x=\frac{-<!--\; -\; -->3b\pm <!--\; \pm \; -->(\sqrt{3(3b^2-<!--\; -\; -->8ac+2a\sqrt[3]{4(2c^3-<!--\; -\; -->9bcd+27a{d}^2+27b^{2}e-<!--\; -\; -->72ace+\sqrt{(2c^3-<!--\; -\; -->9bcd+27a{d}^2+27b^{2}e-<!--\; -\; -->72ace{)}^2-<!--\; -\; -->4(c^2-<!--\; -\; -->3bd+12ae{)}^3})}+2a\sqrt[3]{4(2c^3-<!--\; -\; -->9bcd+27a{d}^2+27b^{2}e-<!--\; -\; -->72ace-<!--\; -\; -->\sqrt{(2c^3-<!--\; -\; -->9bcd+27a{d}^2+27b^{2}e-<!--\; -\; -->72ace{)}^2-<!--\; -\; -->4(c^2-<!--\; -\; -->3bd+12ae{)}^3})})}\pm <!--\; \pm \; -->\sqrt{3(3b^2-<!--\; -\; -->8ac+2a(\frac{-<!--\; -\; -->1+\sqrt{-<!--\; -\; -->3}}{2})\sqrt[3]{4(2c^3-<!--\; -\; -->9bcd+27a{d}^2+27b^{2}e-<!--\; -\; -->72ace+\sqrt{(2c^3-<!--\; -\; -->9bcd+27a{d}^2+27b^{2}e-<!--\; -\; -->72ace{)}^2-<!--\; -\; -->4(c^2-<!--\; -\; -->3bd+12ae{)}^3})}+2a(\frac{-<!--\; -\; -->1-<!--\; -\; -->\sqrt{-<!--\; -\; -->3}}{2})\sqrt[3]{4(2c^3-<!--\; -\; -->9bcd+27a{d}^2+27b^{2}e-<!--\; -\; -->72ace-<!--\; -\; -->\sqrt{(2c^3-<!--\; -\; -->9bcd+27a{d}^2+27b^{2}e-<!--\; -\; -->72ace{)}^2-<!--\; -\; -->4(c^2-<!--\; -\; -->3bd+12ae{)}^3})})})\pm <!--\; \pm \; -->\mathrm{sgn}<!--\; \; -->((\mathrm{sgn}<!--\; \; -->(-<!--\; -\; -->b^3+4abc-<!--\; -\; -->8a^{2}d)-<!--\; -\; -->\frac{1}{2})(\mathrm{sgn}<!--\; \; -->(max((2c^3-<!--\; -\; -->9bcd+27a{d}^2+27b^{2}e-<!--\; -\; -->72ace{)}^2-<!--\; -\; -->4(c^2-<!--\; -\; -->3bd+12ae{)}^3,min(3b^2-<!--\; -\; -->8ac,3b^4+16a^2{c}^2+16a^{2}bd-<!--\; -\; -->16a{b}^{2}c-<!--\; -\; -->64a^{3}e)))-<!--\; -\; -->\frac{1}{2}))\sqrt{3(3b^2-<!--\; -\; -->8ac+2a(\frac{-<!--\; -\; -->1-<!--\; -\; -->\sqrt{-<!--\; -\; -->3}}{2})\sqrt[3]{4(2c^3-<!--\; -\; -->9bcd+27a{d}^2+27b^{2}e-<!--\; -\; -->72ace+\sqrt{(2c^3-<!--\; -\; -->9bcd+27a{d}^2+27b^{2}e-<!--\; -\; -->72ace{)}^2-<!--\; -\; -->4(c^2-<!--\; -\; -->3bd+12ae{)}^3})}+2a(\frac{-<!--\; -\; -->1+\sqrt{-<!--\; -\; -->3}}{2})\sqrt[3]{4(2c^3-<!--\; -\; -->9bcd+27a{d}^2+27b^{2}e-<!--\; -\; -->72ace-<!--\; -\; -->\sqrt{(2c^3-<!--\; -\; -->9bcd+27a{d}^2+27b^{2}e-<!--\; -\; -->72ace{)}^2-<!--\; -\; -->4(c^2-<!--\; -\; -->3bd+12ae{)}^3})})}}{12a}$$

The quartic formula gives the solutions of $a{x}^4+b{x}^3+c{x}^2+dx+e=0$ for real numbers $a$, $b$, $c$, $d$, $e$ with $a\ne <!--\; \ne \; -->0$.

Directions: Choose all possibilities for the three $\pm <!--\; \pm \; -->$ signs with the last two equivalent. Use real cube roots if possible, and principal roots otherwise.