Introduction

• A second degree polynomial has the form y = ax² + bx + c, where a,b, and c are real numbers, x is the independent variable, and y is the dependent variable.
• The graph of equations of these form is a porabola and can have two, one, or no solutions when y = 0.
• The quadratic equation gives us the 'zeroes' for these equations by using the coeficients a,b, and c.

Derivation of the Quadratic Equation

Using algebra we can set y=0 and solve for x in terms of the coeficients a, b, and c:

ax² + bx + c = 0

x² + (b/a)x + c/a = 0

(x + b/2a)² - b² /4a² +c/a = 0

(x + b/2a)² = b² /4a² - c/a

(x + b/2a)² = (b² - 4ac)/4a²

x + b/2a = +/-sqrt((b² - 4ac)/4a²⁾

x = (-b +/-sqrt(b² - 4ac))/2a

An Example

The derivation on the last slide gves us two roots:

• x1 = (-b -sqrt(b² - 4ac))/2a
• x2 = (-b +sqrt(b² - 4ac))/2a

By plugging in the correct coeficients we can find the solution to any second degree polynomial if it exists.

For Example:

2x² + 3x - 1 = 0

a <- 2
b <- 3
c <- -1
x1 <- (-b -sqrt(b^2 - 4*a*c))/(2*a)
x2 <- (-b +sqrt(b^2 - 4*a*c))/(2*a)
y <- function(x){2*x^2+3*x-1}
data.frame("Root" = c(x1,x2), "Value" = c(y(x1),y(x2)))

##         Root Value
## 1 -1.7807764     0
## 2  0.2807764     0

Conclussion

• My shiny app performs these calculations for coeficients input by user
• It uses these calculations to find the zeros for the coresponding second degree polynomial.
• It presents the solutions clearly and provides a graph of the polynomial centered around the zeros to help visualize the solution.
• The ease of use and simplicity of this app makes it ideal for finding the zeros of a second degree polynomial.