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} }}

  • Sum and products of the roots \alpha and \beta of a{x^2} + bx + c = 0 is given by \alpha      + \beta = - \frac{b}{a} and \alpha \beta       = \frac{c}{a}
  • {b^2} - 4ac is called 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 cube root of unity.
  • \omega and {\omega ^2} are called the complex cube root of unity.
  • Each of the complex cube roots of unity is 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}}}

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