Quadratic Formula

The quadratic formula is the name given to a mathematical expression allowing to find the solutions of a quadratic equation (presented in the form of a polynomial of degree 2 equal to 0). This is the easiest method and therefore the most often taught.

For any polynomial $ P $ of order 2, of variable $ x $ denoted $$ P(x) = ax^2+bx+c $$ the solutions of $ P(x) = 0 $ are given by the quadratic formula $$ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} $$

From a polynomial of degree 2, of variable $ x $:

— Expand and reduce the expression of the polynomial if necessary

— Note $ a $ the coefficient associated with $ x^2 $

— Note $ b $ the coefficient associated with $ x $

— Note $ c $ the remaining constant

— Calculate the solutions $$ x_1 = \frac{-b + \sqrt{b^2 - 4ac}}{2a} \\ x_2 = \frac{-b - \sqrt{b^2 - 4ac}}{2a} $$

From the initial equation $$ ax^2+bx+c = 0 $$

— Divide by $ a $: $$ x^2 + \frac{b}{a} x + \frac{c}{a}=0 $$

— Subtract $ \frac{c}{a} $: $$ x^2 + \frac{b}{a} x = -\frac{c}{a} $$

— Apply the square completion, so add $ \left( \frac{b}{2a} \right)^2 $ to get $$ x^2 + \frac{b}{a} x + \left( \frac{b}{2a} \right)^2 = -\frac{c}{a} + \left( \frac{b}{2a} \right)^2 $$

— Simplify $$ \left( x + \frac{b}{2a} \right)^2 = -\frac{c}{a} + \frac{b^2}{4a^2} $$

— Rearrange the fractions on the right over their common denominator $ 4a^2 $: $$ \left( x + \frac{b}{2a} \right)^2 = \frac{b^2-4ac}{4a^2} $$

— Calculate the square root: $$ x + \frac{b}{2a} = \pm \frac{\sqrt{b^2-4ac}}{2a} $$

— Isolate $ x $ to get the final formula: $$ x = \frac{ -b \pm \sqrt{b^2-4ac}}{2a} $$