This is an analytical proof of the quadratic formulas used to solve quadratic equations.

A quadratic equation in the standard form is given by

a x^{2} + b x + c = 0

where a, b and c are constants with a not equal to zero.

Solve the above equation to find the quadratic formulas

Given
a x^{2} + b x + c = 0

Divide all terms by a
x^{2} + b / a x + c / a = 0

Subtract c / a from both sides
x^{2} + (b / a) x + c / a - c/ a = - c / a

and simplify
x^{2} + (b / a) x = - c / a

Add (b / 2a)^{2} to both sides

x^{2} + (b / a) x + (b / 2a)^{2} = - c / a+ (b / 2a)^{2}

to complete the square
(x + b / 2a )^{2} = - c / a + (b / 2a)^{2}

Group the two terms on the right side of the equation
(x + b / 2a)^{2} = (b^{2} - 4 a c) / (4 a^{2} )

Solve by taking the square root
x + b / 2a = ~+mn~ √{ (b^{2} - 4 a c ) / (4 a^{2}) }

Solve for x to obtain two solutions
x = - b / 2a ~+mn~ √{ (b^{2} - 4 a c ) / (4 a^{2}) }

The term √{ (b^{2} - 4 a c ) / (4 a^{2}) }may be simplified as follows
√{ (b^{2} - 4 a c ) / (4 a^{2}) } = √(b^{2} - 4 a c) / (2 |a|)

Since 2 | a | = 2 a for a > 0 and 2 | a | = -2 a for a < 0, the two solutions to the quadratic equation may be written
x = (-b + √( b^{2} - 4 a c)) / (2 a)

x = (-b - √ ( b^{2} - 4 a c)) / (2 a)

The term b^{2} - 4 a c which is under the square root in both solutions is called the discriminant of the quadratic equation. It can be used to determine the number and nature of the solutions of the quadratic equation. 3 cases are possible

case 1: If b^{2} - 4 a c > 0 , the equation has 2 solutions.

case 2: If b^{2} - 4 a c = 0 , the equation has one solutions of mutliplicity 2.

case 3: If b^{2} - 4 a c < 0 , the equation has 2 complex solutions.