To solve the equation x² – 9 – 16 = 0 using the zero product property, we first need to rearrange and simplify the equation.
1. **Rearranging the equation:**
Start by combining the constants:
x² – 9 – 16 = x² – 25 = 0.
Thus, we can rewrite the equation as:
x² – 25 = 0
2. **Recognizing the difference of squares:**
Notice that x² – 25 can be factored as a difference of squares, which has the general form a² – b² = (a – b)(a + b). Here, we identify:
– a = x
– b = 5 (because 5² = 25)
So, we can factor the equation:
(x – 5)(x + 5) = 0
3. **Applying the zero product property:**
The zero product property states that if the product of two factors is zero, then at least one of the factors must be equal to zero. Therefore, we set each factor to zero:
x – 5 = 0 or x + 5 = 0
4. **Solving for x:**
– From x – 5 = 0, we get x = 5.
– From x + 5 = 0, we get x = -5.
5. **Conclusion:**
Therefore, the solutions to the equation x² – 9 – 16 = 0 are:
x = 5 and x = -5.
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