solve P=-¼x²+5x+24 when Your company has a profit that is represented by the equation P= -¼x²+5x+24, where P is the profit in millions and x is the number of years starting in 2018.

To solve for the profit equation P = -¼x² + 5x + 24, we can set P to zero and find the values of x that make the equation true. This will give us the years when the profit would be zero or break-even.

0 = -¼x² + 5x + 24

Multiply the equation by -4 to eliminate the fractions:

0 = x² - 20x - 96

Now, we can try to factor the equation:
0 = (x - 12)(x + 8)

Setting each factor to zero and solving for x:
x - 12 = 0 or x + 8 = 0

x = 12 or x = -8

Since x represents the number of years starting from 2018, we can see that x = 12 corresponds to the year 2030, and x = -8 does not make sense in this context.

Therefore, in the profit equation P = -¼x² + 5x + 24, the profit will be zero or break-even in the year 2030.

To solve the equation P = -¼x² + 5x + 24, we want to find the value(s) of x when the profit P is equal to zero.

Setting P to zero, we have:

0 = -¼x² + 5x + 24

To solve this quadratic equation, we can use the quadratic formula:

x = (-b ± √(b² - 4ac)) / 2a

Here, a = -¼, b = 5, and c = 24.

Substituting these values into the quadratic formula:

x = (-(5) ± √((5)² - 4(-¼)(24))) / (2(-¼))

Simplifying further:

x = (-5 ± √(25 + 24)) / (-½)

x = (-5 ± √49) / (-½)

x = (-5 ± 7) / (-½)

Now let's consider both the positive and negative roots separately:

For the positive root:

x = (-5 + 7) / (-½)

x = 2 / (-½)

x = -4

For the negative root:

x = (-5 - 7) / (-½)

x = -12 / (-½)

x = 24

Therefore, the solution to the equation P = -¼x² + 5x + 24 is x = -4 and x = 24.