Derivation of the Quadratic Formula

The Quadratic Formula is derived by applying the method of completing the square on the standard form of a quadratic equation. Its complete derivation is given below:



  • General form of Quadratic Equation :

  • Standard form of quadratic equation

  • Divide throughout by 'a' (the coefficient of x-squared) :

  • Standard form of quadratic divided throughout by 'a'
    Standard form of quadratic divided throughout by 'a' (result)

  • Move the term
    Fraction c/a
    to the RHS (Right Hand Side) :

  • Moved c/a to the RHS

  • Take the coefficient of 'x', divide it by 2, then square it. Add the resulting term
    to both sides of the quadratic equation :
  • Added (b^2) / (4a^2) to both sides

  • The LHS is now in the form of the expanded product of because
    Expanded expression of (x + b/2a)^2

  • Thus rewriting it in its factored form,
    Factored LHS (equation)
    Simplifying the RHS,
    Factored LHS (equation) with RHS simplified
    Taking square root of both sides,
    Taking square root of both sides
    The square root of the LHS cancels out the exponent of 2 on it; The square root
    of '4a2' in the RHS is '2a', and the square root of the numerator of
    the RHS is either positive or negative.
    Taking square root of both sides (result)
    Move 'b/2a' to the RHS,
    Moved 'b/2a' to the RHS
    The Quadratic Formula (derived),
    which is the Quadratic Formula.
    Thus the Quadratic Formula is derived from the standard form of a quadratic equation
    by applying the method of completing the square on it.

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