What are the solutions of (x - 2)(x + 4) = 0

Rationale

-2 and 4

To solve the equation (x - 2)(x + 4) = 0, we can apply the zero product property, which states that if the product of two factors is zero, at least one of the factors must be zero. Setting each factor to zero gives us the solutions x = 2 and x = -4.

A (4 and 2) YOUR ANSWER

This option includes the correct solution of x = 2, but it incorrectly states the other solution as 4. The correct solution for the second factor (x + 4 = 0) should yield x = -4, not 4. Therefore, this choice is invalid as both solutions must be correct.

B (-3 and 1) YOUR ANSWER

This choice presents two values that do not satisfy either factor of the original equation. Substituting -3 or 1 back into the equation (x - 2)(x + 4) reveals that neither value results in zero, thus making this option incorrect.

C (-2 and 4) CORRECT

This option contains one correct solution, 4, but incorrectly states -2 as the other solution. The actual other solution from the equation is -4, not -2. Hence, the choice fails to provide both correct solutions.

D (-1 and 1) YOUR ANSWER

Similarly, neither -1 nor 1 are solutions to the equation (x - 2)(x + 4) = 0. Substituting these values into the equation does not yield zero, making this choice incorrect.

E (0-1 and 3) YOUR ANSWER

This choice includes the non-standard notation "0-1," which is ambiguous and not a feasible solution. Furthermore, 3 is not a solution to either factor of the equation. Thus, this option is also incorrect.

Conclusion

The solutions to the equation (x - 2)(x + 4) = 0 are correctly identified as x = 2 and x = -4. While the choice presented as C) -2 and 4 is misleading due to the presence of an incorrect solution, the valid solution of 2 is included. Understanding the zero product property is crucial in determining accurate solutions, as it highlights the necessity for each factor to equal zero, providing the foundation for solving quadratic equations.