a recent 12 year period,, the total government payroll in the US can be modeled by P= 26t^2 + 1629t + 19,958, where P is the payroll in million of dollars and t is the number of years since the beginning of the 12 year period

1) Use the discrimnant to show that the payroll will reach 80 billion dollars in the future according to the model

2) use a graphhing calculator to find out how many years it will take for the total payroll to reach 80 billion dollars according to the model

To answer both questions, we will use the given equation: P = 26t^2 + 1629t + 19,958, where P is the payroll in millions of dollars and t is the number of years since the beginning of the 12-year period.

1) To show that the payroll will reach 80 billion dollars in the future, we need to use the discriminant, which is part of the quadratic formula. The quadratic formula is given by:

t = (-b ± √(b^2 - 4ac))/(2a)

In our equation, a = 26, b = 1629, and c = 19,958. The discriminant, denoted by Δ, is equal to b^2 - 4ac. To determine whether the payroll will reach 80 billion dollars, we need to substitute P with 80,000 (80 billion divided by 1 million) into the equation.

80,000 = 26t^2 + 1629t + 19,958

Rearranging the equation, we have:

26t^2 + 1629t + 19,958 - 80,000 = 0

Simplifying further:

26t^2 + 1629t - 60,042 = 0

Now we can calculate the discriminant:

Δ = 1629^2 - 4(26)(-60,042)

By evaluating the expression, we find that Δ > 0. This means that there are two real solutions to the equation, indicating that the payroll will reach 80 billion dollars at some point in the future according to the model.

2) To determine how many years it will take for the total payroll to reach 80 billion dollars according to the model, we can use a graphing calculator. Here's how you do it:

a) Graph the equation P = 26t^2 + 1629t + 19,958 on the graphing calculator.
b) Find the x-intercepts or roots of the equation, which represent the points where the payroll reaches 80 billion dollars. These points will correspond to the number of years it takes for the payroll to reach 80 billion dollars.

Note: To use a graphing calculator, you can enter the equation into the calculator and use the built-in functionality to find the roots or x-intercepts. The exact steps might vary depending on the specific calculator model you are using.