To solve the quadratic equation x² – 7x – 12 = 0 using the zero product property, follow these steps:
- Rearrange the equation in standard form: The equation is already in standard form, which is ax² + bx + c = 0, where a = 1, b = -7, and c = -12.
- Factor the quadratic: We need to factor the quadratic expression on the left side. We are looking for two numbers that multiply to -12 (the value of c) and add up to -7 (the value of b). These two numbers are -3 and +4. Thus, we can factor the equation as follows:
- Set the factors equal to zero: After factoring, we get:
- (x – 3)(x + 4) = 0
- Apply the zero product property: According to the zero product property, 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 equal to zero:
- x – 3 = 0
- x + 4 = 0
- Solve for x: Now we solve each equation:
- From x – 3 = 0, we get x = 3.
- From x + 4 = 0, we get x = -4.
- Conclusion: The solutions to the equation x² – 7x – 12 = 0 are x = 3 and x = -4.