The price C, in dollars per share, of a high-tech stock has fluctuated over a twelve-year period according to the equation C= 14 + 12x - x2, Where x is in years. The price C, in dollars per share, of a second high-tech stock has
shown a steady increase during the same time period according to the relationship C = 2x + 30. For what
values are the two stock prices the same?
We start the problem by setting the two equations for the stock prices equal to each other:
14 + 12x - x^2 = 2x + 30
We can simplify this equation by rearranging and collecting like terms:
-x^2 + 10x - 16 = 0
To solve for x, we can use the quadratic formula:
x = (-b ± sqrt(b^2 - 4ac)) / 2a
In this case, a = -1, b = 10, and c = -16. Plugging in these values, we get:
x = (-10 ± sqrt(10^2 - 4(-1)(-16))) / 2(-1)
Simplifying under the square root:
x = (-10 ± sqrt(100 - 64)) / -2
x = (-10 ± sqrt(36)) / -2
x = (-10 ± 6) / -2
x = 2 or 8
So the two stock prices are the same at x = 2 and x = 8. To find the corresponding prices, we can plug these values back into either equation:
C = 14 + 12(2) - 2^2 = 28
C = 2(2) + 30 = 34
Therefore, the two stock prices are the same at $28 per share and $34 per share, in years 2 and 8 of the twelve-year period.