IMBROGLIO ABSCISSATRON 1.0 - DEVELOPMENT OF THE QUADRATIC FORMULA
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QUADRATIC EQUATION
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Here is the development of the formula for quadratic equations (of the second-degree). This method uses the completion of the square technique (sqrt{} means "the square root of..."):

1. ax + bx + c           = 0
	general form of the quadratic equation

2. ax + bx               = -c
	isolate the constant

3. (ax + bx)/a           = (-c)/a
	divide the equation by the coefficient of the first term 'a'

4. x + (bx)/a            = (-c)/a

5. x + (bx)/a + [b/4a] = (-c)/a + [b/4a]
	addition (b/2a) to the two sides of the equation

6. x + (bx)/a + [b/4a] = (-4ac + b)/4a
	simplifier the expression on the right

7. [x + (b/2a)]          = (-4ac+b)/4a
	factorise the expression on the right

8. sqrt{ [x + (b/2a)] }  = sqrt{ (-4ac + b)/4a }
	find the square roots of the two sides of the equation

9. [x + (b/2a)]           = sqrt{ (-4ac + b)/4a }

10. [x + (b/2a)]           = sqrt{ -4ac+b }/2a

11. x                      = -(b/2a) sqrt{ -4ac+b }/2a
	transport (b/2a) two the right in order to isolate 'x'

12. x                      = (-b sqrt{ b-4ac }/2a
	addition the fractions

We have isolated the variable 'x' from [ax+bx+c=0]

Result: x = (-b  sqrt{ b - 4ac })/2a


The part (b-4ac) from this formula is called the discriminant. It indicates the nature of the roots of the quadratic equation.

d = (b-4ac)

i)   if d > 0 and 'd' is a perfect square:
	the two roots of the equation are different and rational (two x-intercepts)

ii)  if d > 0 and 'd' is not a perfect square:
	the two roots are different and irrational (two x-intercepts)

iii) if d = 0 :
	the two roots are equal (only one x-intercept)

iv)  if d < 0 :
	the roots are irreal (there are no x-intercepts)
	

It's therefore very simple to find the sum and product of the roots of a quadratic equation:

Sum:
	[(-b + sqrt{ b - 4ac })/2a] + [(-b - sqrt{ b - 4ac })/2a]

	the square roots cancel and it it becomes:
	= [-b/(2a)] + [-b/(2a)] = [(-2b)/(2a)] = [-b/a]

	Result: -b/a

Product:
	[(-b + sqrt{ b - 4ac })/2a] * [(-b - sqrt{ b - 4ac })/2a]
	= [(-b + sqrt{ b - 4ac }) * (-b - sqrt{ b - 4ac })]/[4a]
	= [b + b*sqrt{ b - 4ac } - b*sqrt{ b - 4ac } - (b-4ac)]/[4a]
	= [b-b+4ac]/[4a]
	= (4ac)/(4a)
	= c/a

	Result: c/a
	

Summary:
	ax+bx+c=0
	ax+bx=-c
	(ax+bx)/a=-c/a
	x+(bx)/a=-c/a
	x+(bx)/a+b/4a=-c/a+b/4a
	x+(bx)/a+b/4a=(-4ac+b)/4a
	(x+b/2a)=(-4ac+b)/4a
	(x+b/2a)=sqrt((-4ac+b)/4a)
	(x+b/2a)=sqrt(-4ac+b)/2a
	x=-b/2asqrt(-4ac+b)/2a
	quadratic formula: x=[-bsqrt(b-4ac)]/2a

	somme of roots: -b/a
	product of roots: c/a


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QUADRATIC FUNCTION
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The quadratic function, in function of 'y', always forms a parabola.

Each quadratic function can be written in two forms:

i)  general form:		y = ax+bx+c
ii) standard form (canonic):	y = a(x-p)+q
	-> this form allows you to easily find the information necessary to trace the graph of the function

Here is the transformation of a quadratic function from general to standard form (!= means "is not equal to...")

1. y = ax + bx + c

2. y = a[x + (bx)/a] + c
	divide the two first terms by 'a', and isolate 'c'

3. y != a[x + (bx)/a + b/(4a)] + c
	create a perfect square

4. y = a[x + (bx)/a + b/(4a)] + c - a[b/(4a)]
	balance the fonction

5. y = a[x + (b/2a)] + c - a[b/(4a)]
	simplify

6. y = a[x + (b/2a)] + c - [ab/(4a)]

7. y = a[x + (b/2a)] + c - [b/(4a)]

8. y = a[x + (b/2a)] + [(4ac - b)/(4a)]

The values of a, p, and q are:
	a = a
	p = -b/2a
	q = (4ac - b)/(4a)

Here are the definitions of these values in context (how the transforme the parabola):

a ->
	i)   determines the opening of the parabola
		* if a > 0 : the parabola opens upward
		* if a < 0 : the parabola ouvre downward

	ii)  determines the compression of the parabola on the 'y' axis
		* if 0 < a < 1 or 0 > a > -1 : the parabola is compressed on the 'y' axis
		* if a > 1 or a < -1         : the parabola is stretched on the 'y' axis

p ->
	i)   determines the horizonal placement of the parabola (the 'x' value in its vertex is the value of 'p')
	ii)  the equation of the axis of symmetry of the parabola is x = p (one must not forget that in a function such as y = s(x+t) + u, the value of 'p' is '-t'. (One must take the opposite of the second term of the expression multiplied by 'a'.)
	iii) determines what value of 'x' gives the function is maximal or minimal 'y' value

q ->
	i)   determines the vertical placement of the parabola (the value of 'y' in the vertex is the value of 'q')
	ii)  determines either the maximal or minimal value of 'y' (it's a maximum when a < 0, and a minimum when a > 0)

Therefore, the vertex of the parabola is defined by the coordinates (p,q) : vertex = [-b/2a , (4ac - b)/(4a)]

The maximal or minimal value of a quadratic function is: (4ac - b)/(4a)
And this value of 'y' arrives when 'x' is -b/2a

The domaine of the parabola is always defined by: {'x' is a real number}
The co-domaine (image) of the parabola is always defined by: {y >= q} (if a > 0) or {y <= q} (if a < 0)

To find the x-intercepts of a parabola, you replace the value of 'y' with 0, and then solve the quadratic equation.
To find the y-intercept of a parabola, you replace the value of 'x' with 0, and then solve the linear equation.

The coordinates of the point of reflection of a parabola are: [(2p) , (y-intercept)]

Summary:
	y=ax+bx+c
	y=a[x+(bx)/a]+c
	y!=a[x+(bx)/a+b/(4a)]+c
	y=a[x+(bx)/a+b/(4a)]+c-a[b/(4a)]
	y=a[x+(b/2a)]+c-a[b/(4a)]
	y=a[x+(b/2a)]+c-ab/(4a)
	y=a[x+(b/2a)]+c-b/(4a)
	y=a[x+(b/2a)]+(4ac-b)/(4a) -> y=a(x-p)+q

	y = a(x-p)+q
	a = a
	p = b/2a
	q = (4ac-b)/(4a)

	a : compression/opening
	p : horizontal displacement
	q : vertical displacement & maximal/minimal value

	vertex = (p,q) = [-b/2a , (4ac-b)/(4a)]


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INVERSE OF QUADRATIC FUNCTION
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Here's the development of a formula to find the inverse of a quadratic function. (The inverse of the function f(x), f'(x), has the same form as the original function, but has been flipped with respect to the linear function y=x)

1. y = ax + bx + c
	general form of the quadratic function

2. x = ay + by + c
	exchange the 'x' and 'y' values

3. x = a[y + (b/a)y +c/a]
	take out a factor of 'a'

4. x/a = y + (b/a)y + c/a
	divide the equation by the first coefficient 'a'

5. x/a = y + (b/a)y + b/(4a) + c/a - b/(4a)
	create a perfect square and balance the equation

6. x/a + b/(4a) = [y + (b/a)y + b/(4a)] + c/a
	isolate the perfect square on the right

7. x/a + b/(4a) = (y + b/2a) + c/a

8. x/a + b/(4a) - c/a = [y + b/(2a)]

9. x/a + (b-4ac)/(4a) = [y + b/(2a)]
	simplify

10. sqrt[x/a + (b-4ac)/(4a)] = y + b/(2a)
	find the square root of the equation

11. sqrt[x/a + (b-4ac)/(4a)] - b/(2a) = y
	isolate 'y'

The inverse function of 'y = ax + bx + c' is therefore of the form: f-1(x) = sqrt[(x - q)/a] + p
	--> f-1(x) = sqrt{ [x + (b-4ac)/(4a)]/a } - b/(2a)

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