To solve the problem of determining after how many days the two friends will earn the same profit, we need to find the intersection points of the two profit functions, [tex]\( P(x) \)[/tex] and [tex]\( Q(x) \)[/tex].
The profit functions are given as:
[tex]\[ P(x) = -x^2 + 5x + 12 \][/tex]
[tex]\[ Q(x) = 6x \][/tex]
We need to solve the system of equations where [tex]\( P(x) \)[/tex] equals [tex]\( Q(x) \)[/tex].
### Step-by-Step Solution:
1. Set the two equations equal to each other:
[tex]\[ -x^2 + 5x + 12 = 6x \][/tex]
2. Rearrange the equation to set it to zero:
Subtract [tex]\( 6x \)[/tex] from both sides of the equation:
[tex]\[ -x^2 + 5x + 12 - 6x = 0 \][/tex]
Simplify the terms:
[tex]\[ -x^2 - x + 12 = 0 \][/tex]
3. Solve the quadratic equation:
The equation [tex]\( -x^2 - x + 12 = 0 \)[/tex] is a standard quadratic equation in the form [tex]\( ax^2 + bx + c = 0 \)[/tex], where [tex]\( a = -1 \)[/tex], [tex]\( b = -1 \)[/tex], and [tex]\( c = 12 \)[/tex].
4. Apply the quadratic formula:
The quadratic formula is [tex]\( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)[/tex].
Substituting [tex]\( a = -1 \)[/tex], [tex]\( b = -1 \)[/tex], and [tex]\( c = 12 \)[/tex]:
[tex]\[ x = \frac{-(-1) \pm \sqrt{(-1)^2 - 4(-1)(12)}}{2(-1)} \][/tex]
[tex]\[ x = \frac{1 \pm \sqrt{1 + 48}}{-2} \][/tex]
[tex]\[ x = \frac{1 \pm \sqrt{49}}{-2} \][/tex]
[tex]\[ x = \frac{1 \pm 7}{-2} \][/tex]
5. Calculate the solutions:
[tex]\[
x_1 = \frac{1 + 7}{-2} = \frac{8}{-2} = -4
\][/tex]
[tex]\[
x_2 = \frac{1 - 7}{-2} = \frac{-6}{-2} = 3
\][/tex]
6. Determine the viable solution:
We have two solutions: [tex]\( x = -4 \)[/tex] and [tex]\( x = 3 \)[/tex].
Since [tex]\( x \)[/tex] represents the number of days, it must be non-negative. Therefore, [tex]\( x = -4 \)[/tex] is not a viable solution because you cannot have negative days.
The viable solution is [tex]\( x = 3 \)[/tex].
To summarize:
- The viable solution to the question "After how many days will the two students earn the same profit?" is 3 days.
- The nonviable solution is -4 days because negative days are not possible.
This solution process confirms that after 3 days, the two friends will earn the same profit, and the negative solution is not applicable in this context.
