Which point is an x-intercept of the quadratic function f(x) = (x – 8)(x + 9)? (0,8) (0,–8) (9,0) (–9,0)

The correct answer is:  [D]:  " (- 9, 0) " .


Explanation:


Given the quadratic function in  "factored form" ;  

             →  with "y" substituted for:  "f(x)" — as follows:
 

  →   "  y = (x − 8)(x + 9) "  ;   

  →  Find the "x-intercept" of the equation ; 

                 →   {among the answer choices given} ; 


Note:  The "x-intercept(s)" of an equation refer(s) to the coordinates of the point(s) on the graph of the equation at which the graphed equation crosses the "x-axis".

In other words, the "x-intercept(s)" of an equation refer(s) to the solution of the equation at which: " x = 0 " .

At this point, let us consider our given answer choices:

Note:

Consider the first 2 (two) given answer choices:


Choice:  [A]:  " (0, 8) " ; 

Choice:  [B]:  " (0, -8) " .

→  Both of these are INCORRECT ;

→  {since these 2 (two) answer choices have "non-zero" values as
"y-coordinates" .}.  

Note that by definition, all "x-intercepts" MUST have "y-coordinates" with a value of "0" {zero}.
                                       
→  Both of these are INCORRECT ; since these 2 (two) answer choices have "non-zero" values as "y-coordinates".  

→  Note that by definition, all "x-intercepts" MUST have "y-coordinates" with a value of "0" {zero}.  

Note that:  

Choice:  [A]:  " (0, -8)" ; has "- 8" — [ not: "0" ] — as a: "y-coordinate" ;
  
and that :

Choice:  [B]:  " (0, 8) " ; has " 8 " — [not: "0" ] — as a: "y-coordinate".

This narrows our answer choices to the last 2 (two) remaining choices:


Choice:  [C]:  " (9, 0) "  ;  AND: 


Choice:  [D]:  " (- 9, 0) " .


Note that both of them could be "x-intercepts" ; since both of them have
values of "0" {zero} as "y-coordinates" .

→  Let us examining EACH of the remaining 2 (two) answer choices.  It does not matter the order, but let us start with:  Choice:  [C]:  " (9, 0) " .

→  Consider the original equation:

                      →  " y = (x − 8)(x + 9) " 

→  Note the answer choice given for Choice:  [C]:  " (9, 0)" .

→  This means that when we plug in "9" for "x" , we should get "0" for "y" ; 

→  Let us plug in these values for "x" to see if "0" (for "y") holds true:
                   
                      →  0 =?  (9 − 8)(9 + 9) ?? ; 

                      →  0 =? (1) ( 18) ?? ; 
 
                      →  0 ≠  18 ; 

        →    As such:  Choice:  [C]:  " (9, 0)" — is INCORRECT.


→  At this time, we may assume that:  "Choice [D]:  " (-9, 0)" —  the only remaining answer choice is the correct answer.

→  However, we shall examine this "answer choice" appropriately; as follows:


→  Consider the original equation:

   
                   →  " y = (x − 8)(x + 9) " 

→  Note the answer choice for:  [C]:  " ( - 9, 0)" .

→  This means that when we plug in "-9" for "x" , we should get "0" for "y" ; 

→  Let us plug in these values for "x" ;  to see if "0" (for "y") ;  holds true:
                   
                      →  0 =?  (9 − 8)(- 9 + 9) ?? ; 

                      →  0 =? (1) ( 0) ?? ; 
 
                      →  0 =?  0 ?? ; 

                      →  0  = 0  ! Yes!

As such:  

        →  The correct answer is:  Choice:  [D]:  " (-9, 0)" .