What is the improved quadratic formula in graphic form?

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What is the improved quadratic formula in graphic form?

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Answer 1 · 2017-07-09T14:57:19

The improved quadratic formula in graphic form:
$x = - \frac{b}{2 a} \pm \frac{d}{2 a}$ $x = - b/(2a) +- d/(2a)$
$D = d^{2} = b^{2} - 4 a c$ $D = d^2 = b^2 - 4ac$ .

Explanation:

This formula relates the 2 roots of the quadratic equations to the parabola graph of the function y = ax^2 + bx + c.

$x = - \frac{b}{2 a} \pm \frac{d}{2 a}$ $x = - b/(2a) +- d/(2a)$
with $d^{2} = D = b^{2} - 4 a c$ $d^2 = D = b^2 - 4ac$ .
In this formula,
- the quantity $(- \frac{b}{2 a})$ $(-b/(2a))$ represents the x-coordinate of the axis of symmetry.
- The 2 quantities $(\pm \frac{d}{2 a})$ $(+- d/(2a))$ represent the 2distances from the axis of symmetry to the 2 x-intercepts.
Advantages:
- Simpler expression, and easier to remember because students can relate the formula to the parabola graph.
- Simple steps for easier numeric computation.
Example. Solve: $y = 16 x^{2} - 62 x + 21 = 0$ $y = 16x^2 - 62x + 21 = 0$ .
$D = d^{2} = b^{2} - 4 a c = 3844 - 1344 = 2500$ $D = d^2 = b^2 - 4ac = 3844 - 1344 = 2500$ --> $d = \pm 50$ $d = +- 50$
There are 2 real roots:
$x = \frac{62}{32} \pm \frac{50}{32} = \frac{31 \pm 25}{16}$ $x = 62/32 +- 50/32= (31 +- 25)/16$
$x 1 = \frac{56}{16} = \frac{7}{2}$ $x1 = 56/16 = 7/2$ , and $x 2 = \frac{6}{16} = \frac{3}{8}$ $x2 = 6/16 = 3/8$

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What is the improved quadratic formula in graphic form?

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