Question 188303
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Let known <font color=blue>P=Y=Profit or  Losses</font>


Solving for <font color=blue>"x"</font> for number of watches. More importantly to see the X-Intercepts.

Given:{{{P=x^2-13x-80}}}
By Quadratic, where{{{system(a=1,b=(-13),c=(-80))}}}
Discriminant: b^2-4ac= -13^2 - 4(1)(-80)=169+320=489
Then,
{{{x=(-(-13)+-sqrt(489))/(2*1)=(13+-22.11)/2}}}
{{{x=(13+22.11)/2=35.11/2=red(17.55)}}}
{{{x=(13-22.11)/2=-9.11/2=red(-4.55)}}}
X-Intercepts = > <font color=blue>(17.55,0) & (-4.55,0)</font>
For Y-Intercept:
Let fx=0 ----> {{{P=y=0^2-13(0)-80=red(-80)}}}
Y-Intercept = > <font color=blue>(0,-80)</font>


We now get the vertex thru Vertex-Form, {{{a(x-h)^2+k}}}, where{{{system(h=x,k=y)}}}
Going back to our eqn:
{{{P=x^2-13x-80}}}, complete the square, by taking "Half" of the middle constant then "squared":


In our eqn, the middle constant is 13, taking half of it = {{{13/2=6.5}}}, then squared it= {{{6.5^2=highlight(42.25)}}}. We "add" and also "subtract" to the eqn:
{{{P=(x^2-13x+highlight(42.25))highlight(-42.25)-80)}}}
{{{P=(x-6.5)^2-122.25}}} --->follows {{{a(x-h)^2+k}}}
In our eqn, we have vertex, h=<font color=blue>x=6.5</font> & k=<font color=blue>y=-122.25</font>


We see graph,

{{{drawing(500,500,-10,20,-130,30,grid(1),graph(500,500,-10,20,-130,30,x^2-13x-80),blue(circle(17.55,0,.25)),blue(circle(-4.55,0,.25)),blue(circle(0,-80,.25)), blue(circle(6.5,-122.25,.25)))}}}
The Y axis, is where the Profit or Losses. 
So if you have a loss of $50, that equals to <font color=red>-50</font> on the Y axis.


Make a straigth line from <font color=red>-50</font> on the Y axis  to the right until it touches the curve. As it touched the curve, extend the line downwards until it touches the X axis. 


<font color=blue>This will show the number of watches for the loss sale of $50.00.</font>


{{{drawing(500,500,-10,20,-130,30,graph(500,500,-10,20,-130,30,x^2-13x-80),blue(circle(17.55,0,.25)),blue(circle(-4.55,0,.25)),blue(circle(0,-80,.25)),green(line(0,-50,15,-50)),green(line(15,-50,15,0)),red(locate(0.3,-50,-50)),blue(locate(14,6,15=watches)),blue(circle(6.5,-122.25,.25)))}}} ----> <font color=red>15 watches sold</font> for a loss of $50.00


Let's check,
Go back our Eqn, & substitute x=15: 

{{{P=15^2-13(15)-80=225-195-80}}}
{{{highlight(P=-50)}}}, Loss


Thank you,
Jojo</font>