We can often factor a quadratic equation into the product of two binomials. We are then left with an equation of the form (x + d )(x + e) = 0, where d and e are integers.

The zero product property states that, if the product of two quantities is equal to 0, then at least one of the quantities must be equal to zero. Thus, if (x + d )(x + e) = 0, either (x + d )= 0 or (x + e) = 0. Consequently, the two solutions to the equation are x = - d and x = - e.

Example 1: Solve for x: x² - 5x - 14 = 0

x² - 5x - 14 = (x - 7)(x + 2) = 0
x - 7 = 0 or x + 2 = 0
x = 7 or x = - 2

Thus, the solution set is { -2, 7}.

Example 2: Solve for x: x² + 6x + 5 = 0

x² + 6x + 5 = (x + 1)(x + 5) = 0
x + 1 = 0 or x + 5 = 0
x = - 1 or x = - 5

Thus, the solution set is { -1, -5}.

Example 3: Solve for x: 2x² - 16x + 24 = 0

2x² -16x + 24 = 2(x² - 8x + 12) = 2(x - 2)(x - 6) = 0
x - 2 = 0 or x - 6 = 0
x = 2 or x = 6

Thus, the solution set is {2, 6}.

Example 4: Solve for x: x² + 6x + 9 = 0

x² +6x + 9 = (x + 3)(x + 3) = (x + 3)² = 0
x + 3 = 0
x = - 3

Thus, the solution set is { -3}.