The number of horses occupying stalls at the county fairgrounds can be modeled by the following function, where x represents the number of days since the first of june. h(x)=-3x^2+30+225 rewrite this function in the form that reveals when there are no horses occupying stalls at the fairgrounds.

 

The rewritten equation is:  

       " - 3x² + 30 + 225 = 0 " ; 

which is ultimately simplified to:

        " x² − 10 − 75  = 0 " .

The number of days is:  "15 days" . 


Explanation:

Given:

    h(x) = - 3x² + 30 + 225  ;  

Rewrite as:  "  - 3x² + 30 + 225 = 0 " ; 

Multiply EACH SIDE of the equation by "-1" ; to get rid of the "-3" ; 

  -1 * {- 3x² + 30 + 225 = 0} ; 

to get:  " 3x² − 30 − 225 = 0" ; 

Divide EACH SIDE of the equation by "3" ; to simplify; since:

"3", "30", and "225" are divisible by "3" ; 

{3x² − 30 − 225} / 3 = 0 / 3 ; 

 to get: 

 " x² − 10 − 75  = 0 " ; 
 

 Let us factor:

-15 + 5 = -10 ; 

-15 * 5 = -75 ; 

Rewrite as:

(x − 15) (x + 5) =  0 ;

x = 15;  and x = -5 ; 

The number of days cannot be "negative value";  so we stick with:

"x = 15 days" 

 The rewritten equation is:  

       " - 3x² + 30 + 225 = 0 " ; 

which is ultimately simplified to:

        " x² − 10 − 75  = 0 " .

The number of days is:  "15 days" .