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

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