The price of a certain stock, in dollars, can be modeled by the function f(x)=0.75x2−6x+20

Question

, where x equals the number of months that the stock is on the exchange. Over the same 6-month period, the price of another stock, in dollars, increases according to the function g(x)=2.75+1.50
. Graph the functions in the same coordinate plane to determine when f(x)=g(x)
. How many times during the 6-month period will both stocks be at the same price?(1 point)
Responses

never
never

once
once

three times
three times

twice

Answer 1

To find when the two functions \( f(x) = 0.75x^2 - 6x + 20 \) and \( g(x) = 2.75 + 1.50 \) are equal, we first need to simplify the linear function \( g(x) \).

The function \( g(x) \) simplifies to:

\[ g(x) = 4.25 \]

Now we need to set \( f(x) \) equal to \( g(x) \) and solve for \( x \):

\[ 0.75x^2 - 6x + 20 = 4.25 \]

Next, we subtract 4.25 from both sides:

\[ 0.75x^2 - 6x + 15.75 = 0 \]

To simplify this equation, we can multiply by 4 (to eliminate the decimal):

\[ 3x^2 - 24x + 63 = 0 \]

Next, we can apply the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), where \( a = 3, b = -24, c = 63 \).

Calculating the discriminant:

\[ b^2 - 4ac = (-24)^2 - 4(3)(63) = 576 - 756 = -180 \]

Since the discriminant is negative, this means that the quadratic equation has no real solutions. Therefore, the graphs of \( f(x) \) and \( g(x) \) do not intersect.

Thus, the answer to the question of how many times during the 6-month period will both stocks be at the same price is:

never.