SOLUTION: If P(x) =5+8x-5x^2 represents the profit in selling x thousand Boombotix speakers, how many speakers should be sold to maximize profit?

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Question 1100684: If P(x) =5+8x-5x^2 represents the profit in selling x thousand Boombotix speakers, how many speakers should be sold to maximize profit?
Found 2 solutions by josmiceli, ikleyn:
Answer by josmiceli(19441) About Me  (Show Source):
You can put this solution on YOUR website!
+P%28x%29+=+-5x%5E2+%2B+8x+%2B+5+
The x-value of the peak ( max profit ) is:
+x%5Bmax%5D+=+-b%2F%282a%29+
+a+=+-5+
+b+=+8+
----------------
+x%5Bmax%5D+=+-8%2F%282%2A%28-5%29%29+
+x%5Bmax%5D+=+8%2F10+
+.8%2A1000+=+800+
800 speakers should be sold
------------------------------
check:
Here's the plot
Plug +x+=+.8+ back into equation to get +P%5Bmax%5D+
+graph%28+400%2C+400%2C+-1%2C+2%2C+-1%2C+10%2C+-5x%5E2+%2B+8x+%2B+5+%29+

Answer by ikleyn(52812) About Me  (Show Source):
You can put this solution on YOUR website!
.
The core knowledge for solving such problems IS THIS 

    If you are given a quadratic function y = ax^2 + bx + c,

    then it gets its maximum/minimum at x= -b%2F2a.

In your case, a= -5, b= 8 and c= -5, so the maximum is achieved at x = %28-8%29%2F%282%2A%28-5%29%29 = 8%2F10 of thousand = 800 speakers.


To read and learn more about this subject, see the lessons
    - HOW TO complete the square to find the minimum/maximum of a quadratic function
    - Briefly on finding the minimum/maximum of a quadratic function
    - HOW TO complete the square to find the vertex of a parabola
    - Briefly on finding the vertex of a parabola

    - A rectangle with a given perimeter which has the maximal area is a square

    - A farmer planning to fence a rectangular garden to enclose the maximal area
    - A farmer planning to fence a rectangular area along the river to enclose the maximal area
    - A rancher planning to fence two adjacent rectangular corrals to enclose the maximal area
    - Using quadratic functions to solve problems on maximizing revenue/profit

    - OVERVIEW of lessons on finding the maximum/minimum of a quadratic function


Also,  you have this free of charge online textbook in ALGEBRA-I in this site
    - ALGEBRA-I - YOUR ONLINE TEXTBOOK.

The referred lessons are the part of this textbook under the topic "Finding minimum/maximum of quadratic functions".


Save the link to this online textbook together with its description

Free of charge online textbook in ALGEBRA-I
https://www.algebra.com/algebra/homework/quadratic/lessons/ALGEBRA-I-YOUR-ONLINE-TEXTBOOK.lesson

to your archive and use it when it is needed.