Jacobcon2idn
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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.

Sagot :

Semsee45idn

Answer:

System of Equations

[tex]\large \begin{cases}P(x)=-x^2+7x+10\\Q(x)= 4x\end{cases}[/tex]

Use the substitution method to solve the given system of equations:

[tex]\large\begin{aligned}P(x) & = Q(x)\\-x^2+7x+10 & = 4x\\-x^2+7x+10 - 4x & = 0\\-x^2+3x+10 & = 0\\x^2-3x-10 & = 0\\x^2-5x+2x-10 & = 0\\x(x-5)+2(x-5) & = 0\\(x+2)(x-5) & = 0\\\implies x+2 & = 0 \implies x=-2\\\implies x-5 & = 0 \implies x=5\end{aligned}[/tex]

As [tex]x[/tex] represents the number of days, then [tex]x\geq 0[/tex]

Therefore:

  • The viable answer is when  [tex]x=5[/tex]   →   5 days
  • The non-viable answer is when  [tex]x=-2[/tex]  (as the number of days cannot be negative)

So the two students will earn the same profit after 5 days.

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