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### 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 :

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

• Move the term
to the RHS (Right Hand Side) :

• Take the coefficient of 'x', divide it by 2, then square it. Add the resulting term
to both sides of the quadratic equation :

• The LHS is now in the form of the expanded product of because

• Thus rewriting it in its factored form,
Simplifying the RHS,
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.

Move 'b/2a' to the RHS,

,