Question

Two friends wash cars to make extra money. The profit P(x) of one friend after x days can be represented by the function P(x) = −x^2 + 7x + 10. The second friend's profit can be determined by the function Q(x) = 4x. Solve the system of equations. What solution is a viable answer to the question, "After how many days will the two students earn the same profit?" and which solution is a nonviable answer? Show your work and justify your answer.

Answer 1

After 5 days, the two friends will have earned the same profit.

To find when the two friends will earn the same profit, you need to set the profit functions P(x) and Q(x) equal to each other and solve for x:

P(x)=Q(x)

Given that

P(x)=−x^2 +7x+10 and Q(x)=4x, the equation becomes:

−x^2 +7x+10=4x

Now, let's solve for x:

−x^2+7x+10−4x=0

Combine like terms:

−x^2 +3x+10=0

Now, to solve this quadratic equation, you can use factoring, completing the square, or the quadratic formula. In this case, I'll use the quadratic formula:

x= −b± b^2 −4ac/ 2a

​Here,

a=−1,

b=3, and

c=10.

Plug these values into the formula:

x= −3± 3^2−4(−1)(10)/ 2(−1)

​Simplify:

x= −3± 9+40/−2

x= −3±49/−2

x= −3±7/−2

This gives two possible solutions:

x= 4/−2 =−2

x= −10/−2 =5

Now, we need to consider the viability of each solution.

The nonviable answer is x=−2 because the number of days cannot be negative in this context.

The viable answer is x=5, and this represents the number of days after which the two friends will earn the same profit.

So, after 5 days, the two friends will have earned the same profit.