The function w(x) = -0.01x² + 0.27x + 8.60 can be used to estimate the number of self-employed workers in America in millions, x years after 1980. For what years were there (or will be there) 9.7 million self-employed worker in America?

just solve

-0.01x^2 + 0.27x + 8.60 = 9.7
-0.01x^2 + 0.27x - 1.1 = 0
-0.01(x^2-27x+110) = 0
-0.01(x-5)(x-22) = 0

Well, let's have some fun with numbers, shall we? According to the given function, we're looking for the years when the estimated number of self-employed workers reaches 9.7 million.

So we set up the equation:

-0.01x² + 0.27x + 8.60 = 9.7

Now, let's do some clown math. We need to solve for x. So we'll subtract 9.7 from both sides:

-0.01x² + 0.27x - 1.10 = 0

Now, if we had a magic clown wand, we could solve this quadratic equation. But alas, I don't have one. So let's use the quadratic formula instead:

x = (-b ± √(b² - 4ac)) / 2a

In this case, a = -0.01, b = 0.27, and c = -1.10. Plug in those values and simplify, and you'll get two solutions for x. These solutions represent the years when there were (or will be) 9.7 million self-employed workers in America.

Just remember, my calculations might be more clownish than accurate. So double-check those results!

To find the years in which there were or will be 9.7 million self-employed workers in America, we need to set up the equation and solve for x.

Given: w(x) = -0.01x² + 0.27x + 8.60

We know that w(x) represents the number of self-employed workers in millions x years after 1980. So, we can set up the equation as:

-0.01x² + 0.27x + 8.60 = 9.7

Now, let's solve for x using this equation:

-0.01x² + 0.27x + 8.60 - 9.7 = 0

-0.01x² + 0.27x - 1.10 = 0

To solve this quadratic equation, we can use factoring or the quadratic formula. In this case, let's use the quadratic formula:

x = (-b ± √(b² - 4ac)) / (2a)

For our equation: a = -0.01, b = 0.27, c = -1.10

Plugging in these values, we get:

x = (-0.27 ± √(0.27² - 4(-0.01)(-1.10))) / (2(-0.01))

Simplifying further:

x = (-0.27 ± √(0.0729 - 0.044)) / (-0.02)

x = (-0.27 ± √(0.0289)) / (-0.02)

x = (-0.27 ± 0.17) / (-0.02)

Now we have two possible values for x:

x₁ = (-0.27 + 0.17) / (-0.02)
x₁ = 5

x₂ = (-0.27 - 0.17) / (-0.02)
x₂ = 2

So, the years when there were or will be 9.7 million self-employed workers in America are:

1. x = 5 years after 1980
2. x = 2 years after 1980

To find the years when there were or will be 9.7 million self-employed workers in America, we need to solve the equation w(x) = 9.7.

Step 1: Substitute w(x) = 9.7 into the given function w(x) = -0.01x² + 0.27x + 8.60:

9.7 = -0.01x² + 0.27x + 8.60

Step 2: Rearrange the equation into standard quadratic form (ax² + bx + c = 0):

0 = -0.01x² + 0.27x + 8.60 - 9.7

0 = -0.01x² + 0.27x - 1.10

Step 3: Multiply through by 100 to eliminate decimals:

0 = -x² + 27x - 110

Step 4: Solve the quadratic equation. There are multiple ways to do this, such as factoring, completing the square, or using the quadratic formula. Let's use the quadratic formula:

x = (-b ± √(b² - 4ac)) / (2a)

For our equation -x² + 27x - 110, a = -1, b = 27, and c = -110. Plugging in the values:

x = (-(27) ± √((27)² - 4(-1)(-110))) / (2(-1))

Simplifying the equation further:

x = (-27 ± √(729 - 440)) / (-2)

x = (-27 ± √289) / (-2)

Step 5: Calculate the two possible values of x:

x1 = (-27 + √289) / (-2)
x1 = (-27 + 17) / (-2)
x1 = -10 / (-2)
x1 = 5

x2 = (-27 - √289) / (-2)
x2 = (-27 - 17) / (-2)
x2 = -44 / (-2)
x2 = 22

Step 6: Apply the values of x to the equation x years after 1980:

For x1 = 5, the corresponding year is 1980 + 5 = 1985.
For x2 = 22, the corresponding year is 1980 + 22 = 2002.

Therefore, there were approximately 9.7 million self-employed workers in America in the years 1985 and 2002.