What is the quadratic formula for solving ax² + bx + c = 0?

The quadratic formula is derived from the process of completing the square on a general quadratic equation of the form ( ax² + bx + c = 0 ). The goal is to isolate ( x ) and determine its possible values based on the coefficients ( a ), ( b ), and ( c ).

Starting from the standard form, when we apply completing the square, we rearrange and manipulate the equation, which ultimately leads us to the formula:

[

x = \frac{-b \pm \sqrt{b² - 4ac}}{2a}

]

In this formula, ( -b ) represents the opposite sign of the linear coefficient, ( b ), while ( \sqrt{b² - 4ac} ) is the square root of the discriminant, which determines the nature of the roots. The ( \pm ) symbol indicates that there can be two possible solutions for ( x ), corresponding to the two values that can be produced by adding or subtracting the square root. The denominator ( 2a ) scales the entire expression according to the leading coefficient of the quadratic term.

This formula effectively gives us the roots of the quadratic equation, revealing both real and complex solutions depending