Question

What is the improved quadratic formula in graphic form?

Answer 1

The improved quadratic formula in graphic form:
`x = - b/(2a) +- d/(2a)`
`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 = - b/(2a) +- d/(2a)`
with `d^2 = D = b^2 - 4ac`.
In this formula,
- the quantity `(-b/(2a))` represents the x-coordinate of the axis of symmetry.
- The 2 quantities `(+- 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 = 16x^2 - 62x + 21 = 0`.
`D = d^2 = b^2 - 4ac = 3844 - 1344 = 2500` --> `d = +- 50`
There are 2 real roots:
`x = 62/32 +- 50/32= (31 +- 25)/16`
`x1 = 56/16 = 7/2`, and `x2 = 6/16 = 3/8`