Lesson Lesson Title: The quadratic function and quadratic formula in general intercept form

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Generalities.

In general, we can convert any quadratic function, in standard form y = ax^2 + bx + c, into the intercept form: y = a*(x - x1)(x - x2), where x1 and x2 are the 2 x-intercepts of the parabola graph.

The quadratic function in intercept form.

y = a*(x^2 + bx/a + c/a) (1). Recall the development of the quadratic formula.
(x^2 + bx/a + c/a)+ (b^2/4a^2 - b^2/4a^2) = (x + b/2a)^2 - (b^2 - 4ac)/4a^2.
If we call d^2 = b^2 - 4ac (2), then we get the general intercept form of the quadratic function:
y = a*(x + b/2a + d/2a)(x + b/2a - d/2a) (3)
This is a general form. When the values of a, b, c, and d are substituted into the expression (3), we will find the specific intercept form of the quadratic function.

Example 1: Convert the equation y = 3x^2 + 16x - 12 into intercept form
Solution. First find d by relation (2)--> d^2 = b^2 - 4ac = 256 + 144 = 400 --> d = 20 and d = -20.
Replace values of d, a, b, c into expression (3), we get the intercept form
y = 3(x + 6)(x - 2/3) = (x + 6)(3x - 2)

The Quadratic Formula in Intercept Form.


From expression (3), we get the new Quadratic Formula in Intercept Form:
x = -b/2a + (or -)d/2a. (4)
In this formula, d will be computed from relation (2)--> d^2 = b^2 - 4ac
- The quantity (-b/2a) represents the x-coordinate of the parabola axis.
- The 2 quantities (-d/2a) and (d/2a)represent the 2 equal distances from the parabola axis to the two x-intercepts.
- If d^2 = 0, there is double root at x = -b/2a
- If d^2 > 0, there are 2 x-intercepts (2 real roots). d may be whole numbers or radical.
- If d^2 < 0, there are no intercepts. d is imaginary. There are 2 complex roots.
Example 2. Solve 7x^2 + 18x - 25 = 0.
Solution. First find d by the relation (2) --> d^2 = 324 + 700 = 1024 --> d = 32 and d = -32.
The formula in intercept form (4) gives the 2 real roots:
x1 = -18/14 + 32/14 = 14/14 = 1
x2 = -18/14 - 32/14 = -50/14 = -25/7.
Example 3. Solve: 5x^2 - 10x - 3 = 0.
Solution. Find d^2 = 100 + 60 = 160 --> d = 12.65 and d = -12.65.
The formula in intercept form (4) gives the 2 real roots:
x1 = 10/10 + 12.65/10 = 22.65/10 = 2.26
x2 = 10/10 - 12.65/10 = -2.65/10 = -0.26
[This lesson was written by Nghi H Nguyen, the author of the new Transforming Method (see Algebra.com)for solving quadratic equations - Updated on Nov 21, 2014]
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