To solve the equation x² – 13x – 30 = 0 using the zero product property, we first need to factor the quadratic expression on the left side of the equation.
The zero product property states that if the product of two factors equals zero, then at least one of the factors must also be zero. Thus, we can express the quadratic equation in factored form: (x – a)(x – b) = 0, where a and b are the solutions we need to find.
1. **Factoring the Quadratic**:
We need to find two numbers that multiply to -30 (the constant term) and add up to -13 (the coefficient of the x term). After testing several pairs of factors, we find that:
- -15 and +2
These numbers work because:
- -15 × 2 = -30
- -15 + 2 = -13
So, we can write the factored form of the equation as:
(x – 15)(x + 2) = 0
2. **Applying the Zero Product Property**: Now, we set each factor equal to zero:
- x – 15 = 0 leads to x = 15
- x + 2 = 0 leads to x = -2
3. **Conclusion**: The solutions to the equation x² – 13x – 30 = 0 are:
- x = 15
- x = -2
Thus, we have successfully used the zero product property to find the solutions!