To determine the number of days after which both friends earn the same profit, we need to solve the system of equations where the profit functions [tex]\(P(x)\)[/tex] and [tex]\(Q(x)\)[/tex] are equal.
The profit functions are given as:
[tex]\[ P(x) = -x^2 + 5x + 12 \][/tex]
[tex]\[ Q(x) = 6x \][/tex]
We set these two profit functions equal to each other to find the point where both friends have the same profit:
[tex]\[ -x^2 + 5x + 12 = 6x \][/tex]
First, we need to set up the equation by moving all terms to one side to form a quadratic equation:
[tex]\[ -x^2 + 5x + 12 - 6x = 0 \][/tex]
[tex]\[ -x^2 - x + 12 = 0 \][/tex]
Next, we solve the quadratic equation [tex]\( -x^2 - x + 12 = 0 \)[/tex]. We start by using the quadratic formula, which is given by:
[tex]\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \][/tex]
Here, [tex]\( a = -1 \)[/tex], [tex]\( b = -1 \)[/tex], and [tex]\( c = 12 \)[/tex].
Substitute the values into the quadratic formula:
[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]
From this, we get two solutions:
[tex]\[ x = \frac{1 + 7}{-2} \][/tex]
[tex]\[ x = \frac{8}{-2} \][/tex]
[tex]\[ x = -4 \][/tex]
and
[tex]\[ x = \frac{1 - 7}{-2} \][/tex]
[tex]\[ x = \frac{-6}{-2} \][/tex]
[tex]\[ x = 3 \][/tex]
Thus, the two solutions are [tex]\( x = -4 \)[/tex] and [tex]\( x = 3 \)[/tex].
Now, we need to determine the viability of each solution by checking if the profits are non-negative for both friends:
1. For [tex]\( x = 3 \)[/tex]:
[tex]\[ P(3) = -3^2 + 5(3) + 12 \][/tex]
[tex]\[ P(3) = -9 + 15 + 12 \][/tex]
[tex]\[ P(3) = 18 \][/tex]
[tex]\[ Q(3) = 6(3) \][/tex]
[tex]\[ Q(3) = 18 \][/tex]
Both [tex]\( P(3) \)[/tex] and [tex]\( Q(3) \)[/tex] are non-negative and equal. Hence, this solution is viable.
2. For [tex]\( x = -4 \)[/tex]:
[tex]\[ P(-4) = -(-4)^2 + 5(-4) + 12 \][/tex]
[tex]\[ P(-4) = -16 - 20 + 12 \][/tex]
[tex]\[ P(-4) = -24 \][/tex]
[tex]\[ Q(-4) = 6(-4) \][/tex]
[tex]\[ Q(-4) = -24 \][/tex]
Both [tex]\( P(-4) \)[/tex] and [tex]\( Q(-4) \)[/tex] are negative. Hence, this solution is nonviable.
In conclusion, the viable solution, i.e., the number of days after which both friends will earn the same profit, is [tex]\( x = 3 \)[/tex]. The nonviable solution is [tex]\( x = -4 \)[/tex] as it leads to negative profits for both friends.
