Question

determine the ( x )-intercepts of the following equation.

( (x - 3)(x + 5) = y )

answer

( (0,15) )
( (-15,0) )
( (3,0) ) and ( (-5,0) )
( (0,3) ) and ( (0,-5) )
( (3,0) ) and ( (5,0) )
( (0,-15) )

Explanation:

Step1: Recall x-intercept definition

x-intercepts occur where \( y = 0 \). So set \( y = 0 \) in the equation \( (x - 3)(x + 5)=y \).
Expression: \( (x - 3)(x + 5)=0 \)

Step2: Solve for x using zero - product property

The zero - product property states that if \( ab = 0 \), then either \( a = 0 \) or \( b = 0 \).
For \( (x - 3)(x + 5)=0 \), we have two cases:
Case 1: \( x - 3=0 \), then \( x = 3 \).
Case 2: \( x + 5=0 \), then \( x=- 5 \).
When \( x = 3 \), \( y = 0 \), so the point is \( (3,0) \). When \( x=-5 \), \( y = 0 \), so the point is \( (-5,0) \).

Answer:

\( (3,0) \) and \( (-5,0) \)