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