What is the improved quadratic formula in graphic form?

Jul 4, 2017

$x = - \frac{b}{2 a} \pm \frac{d}{2 a}$
$D = {d}^{2} = {b}^{2} - 4 a c$

Explanation:

$x = - \frac{b}{2 a} \pm \frac{d}{2 a}$,
$D = {d}^{2} = {b}^{2} - 4 a c$.

a, b, and c are the coefficients of the quadratic equation,
$- \frac{b}{2 a}$ is the coordinate of the axis of symmetry, or of the vertex
(+- d/2a) are the distances from the axis of symmetry to the 2 x-intercepts.
Example. Solve: $8 {x}^{2} - 22 x - 13 = 0$
$D = {d}^{2} = {b}^{2} - 4 a c = 484 + 416 = 900$ --> $d = \pm 30$
There are 2 real roots:
$x = - \frac{b}{2 a} \pm \frac{d}{2 a} = \frac{22}{16} \pm \frac{30}{16} = \frac{11 \pm 15}{8}$
$x 1 = \frac{16}{8} = 2$
$x 2 = - \frac{4}{8} = - \frac{1}{2}$