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# 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 |

vCalc.2nd Degree Polynomial

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:

- `a*x^2`
- `b*x`
- `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 **c **can also be zero. Technically, the **a **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`

**2nd Degree Polynomial**, is used in 2 calculators.

**Calculators**