2nd Degree Polynomial

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"Result" =
2nd Degree Polynomial
Variable Instructions Datatype
(a)"Coefficient of " x^2 Enter a real number for the multiplier of the x-squared term Decimal
(b)"Coefficient of " x Enter a real number for the coefficient of the x term Decimal
(c)"Constant Term" Enter a real number of the constant term Decimal
(x)"Value of independent x" Enter a real number to be substituted for x Decimal
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ID
vCalc.2nd Degree Polynomial
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e6cc5fd5-da27-11e2-8e97-bc764e04d25f

A second degree polynomial is where a, b, and c are each multiplicative constants and x is the independent variable with powers of zero (constant), one and two (squared).

INPUTS

This equation has the following inputs:

• a - the multiplicative constant for the x^2 term
• b - the multiplicative constant for the x term
• c - constant term
• x - the value to be substituted for x in both the the x^2 term and the x term

DERIVATION

A second degree polynomial is an equation of the form:

y = ax^2 + bx + c    (showing the multiplications explicitly: y = a*x^2 + b*x + c)

In this simple algebraic form there are three additive terms shown on the right of the equation:

1.    a*x^2
2.    b*x
3.   c

The equation is called "second degree" because it has an x^2 term.  If it also had an x^3 term in it, it would be called a third degree polynomial.

Note that the three terms could be thought of as each term having a multiplying constant: a, b, and c

• where a is a constant multiplying the x^2 term
• where b is a constant multiplying the x^1 term (x to the first power is just x)
• where c is a constant multiplying the x^0 term (x to the zero power is just 1)

So, the form of the second degree polynomial could be re-written more explicitly to be:

y = a*x^2 + b*x^1 + c*x^0

EXAMPLES

We can make up examples values of a, b, and c to form an infinite number of variations on a second degree polynomial.  The multiplicative constants,  a, b, and c, can have integer or decimal values:

ex. y = 5x^2 + 7x + 23  ;  and if x = 3, y = 89

ex. y = 3.7x^2 + 0.4x + 22.2 ; and if x = 6.5, y = 181.125

The multiplicative constants for b and can also be zero.  Technically, the constant can't be zero if the equation is to be called a "second degree polynomial" but this brings up an interesting point: a lower degree polynomial is only a higher degree polynomial with all the higher order terms having multiplicative constants equal to zero. a*x^2 + b*x + c = 0*x^5 + 0*x^4 + 0*x^3 + a*x^2 + b*x + c

Here are two examples of the b and the c constants being zero.

•       ex. y = 5x^2 + 0x + 23 = 5x^2 + 23
•       ex. y = 2x^2 + 6.5x + 0 = 2x^2 + 6.5x

Here is an example of one of the constants, the b constant, being a negative multiplier:

•      ex. y = 3.7x^2 - 1.4x + 22.2
•      ex. y = 7x^2 - 4x + 2