Final Answer:
The solutions to the equation (7+z)(4z-6)=0 are z=-7 and z=3/2.
The equation is solved by setting each factor equal to zero resulting in the solutions z=-7 and z=3/2 according to the zero-product property.
Step-by-step explanation:
The given equation (7+z)(4z-6)=0 involves two factors whose product equals zero. According to the zero product property this implies that at least one of the factors must be zero. Therefore we set each factor separately to zero and solve for the variable z.
Firstly setting 7+z equal to zero, we find z = -7. This indicates one solution to the original equation. Secondly setting 4z-6 equal to zero we find z= 3/2 providing a second solution.
In essence the solutions z = -7 and z = 3/2 signify the points at which either of the two factors becomes zero making the entire product zero. This principle aligns with the fundamental concept that the product of any quantity and zero is always zero.
Consequently understanding and applying the zero product property proves crucial in algebraic problem solving particularly when dealing with quadratic equations and factored expressions.