# Results and Formulas of Equations

• If $\alpha$ and $\beta$ are the roots of the Quadratic Equation $a{x^2} + bx + c = 0$, then

$\alpha = \frac{{ – b + \sqrt {{b^2} – 4ac} }}{{2a}}\,and\,\beta = \frac{{ – b – \sqrt {{b^2} – 4ac} }}{{2a}}$

$\alpha = \frac{{ – 2c}}{{b + \sqrt {{b^2} – 4ac} }}\,and\,\beta = \frac{{ – 2c}}{{b – \sqrt {{b^2} – 4ac} }}$

• The sum and products of the roots $\alpha$ and $\beta$ of $a{x^2} + bx + c = 0$ are given by $\alpha + \beta = – \frac{b}{a}$ and $\alpha \beta = \frac{c}{a}$
• ${b^2} – 4ac$ is called the discriminant of $a{x^2} + bx + c = 0$
• The roots of the quadratic $a{x^2} + bx + c = 0$ are
• imaginary if ${b^2} – 4ac$ is negative.
• real if ${b^2} – 4ac$ is positive or zero.
• real and equal if ${b^2} – 4ac = 0$.
• real and rational if ${b^2} – 4ac \geqslant 0$ and ${b^2} – 4ac$ is a perfect square or zero.
• real and irrational if ${b^2} – 4ac > 0$ and ${b^2} – 4ac$ is not a perfect square.
• The equation whose roots are $\alpha$ and $\beta$ (given) is given by ${x^2} – (\alpha + \beta )x + \alpha \beta = 0$
• $1$, $\omega$ and ${\omega ^2}$ where $\omega = \frac{{ – 1 + i\sqrt 3 }}{2}$ and ${\omega ^2} = \frac{{ – 1 – i\sqrt 3 }}{2}$ are called the cube root of unity.
• $\omega$ and ${\omega ^2}$ are called the complex cube root of unity.
• Each of the complex cube roots of unity are the square of the other.
• The sum of the cube roots of unity is zero. i.e.,$1 + \omega + {\omega ^2} = 0$
• ${\omega ^3} = 1$
• If $\alpha$,$\beta$ and $\gamma$ are the roots of ${a_0}{x^3} + {a_1}{x^2} + {a_2}x + {a_3} = 0$ then

$\alpha + \beta + \gamma = – \frac{{{a_1}}}{{{a_0}}}$
$\alpha \beta + \beta \gamma + \alpha \gamma = \frac{{{a_2}}}{{{a_0}}}$
$\alpha \beta \gamma = – \frac{{{a_3}}}{{{a_0}}}$

• If $\alpha$,$\beta$,$\gamma$ and $\delta$ are the roots of ${a_0}{x^4} + {a_1}{x^3} + {a_2}{x^2} + {a_3}x + {a_4} = 0$ then

$\alpha + \beta + \gamma + \delta = – \frac{{{a_1}}}{{{a_0}}}$
$\alpha \beta + \alpha \gamma + \alpha \delta + \beta \gamma + \beta \delta + \gamma \delta = \frac{{{a_2}}}{{{a_0}}}$
$\alpha \beta \gamma + \alpha \beta \delta + \alpha \gamma \delta + \beta \gamma \delta = – \frac{{{a_3}}}{{{a_0}}}$
$\alpha \beta \gamma \delta = \frac{{{a_4}}}{{{a_0}}}$