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Solving Using the Zero Product Property
Solve by applying the Zero Product Property.
Step 1 — Set each factor equal to zero:
Step 2 — Solve each linear equation. Add/subtract the constant term, then divide by the coefficient:
Step 3 — Check by substituting each value into the original equation: For : \left(4\left(-\frac{3}{4} ight) + 3 ight)\left(4\left(-\frac{3}{4} ight) - 3 ight) = (-3 + 3)(-3 - 3) = 0 \cdot (-6) = 0 ✓ For : \left(4\left(\frac{3}{4} ight) + 3 ight)\left(4\left(\frac{3}{4} ight) - 3 ight) = (3 + 3)(3 - 3) = 6 \cdot 0 = 0 ✓
The solutions are and .
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Intermediate Algebra @ OpenStax
Ch.6 Factoring - Intermediate Algebra @ OpenStax
Algebra
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In a corporate financial spreadsheet, the total revenue for a service is calculated as the product of the number of clients and the service fee. If the total revenue is recorded as $0, which statement correctly applies the Zero Product Property?
In a corporate inventory system, the 'Total Value' of a product is calculated by multiplying the 'Quantity' by the 'Unit Cost'. If the Total Value is $0, the Zero Product Property states that either the Quantity must be 0 or the ____ must be 0.
In a corporate quality control process, the 'Total Defect Impact' is calculated by multiplying the 'Number of Defects' by the 'Cost per Defect'. According to the Zero Product Property, if the Total Defect Impact is 0, then it must be true that at least one of the factors (Number of Defects or Cost per Defect) is 0.
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In professional operations, financial and production outcomes are often calculated by multiplying two or more variables. To recall the logic of the Zero Product Property, match each mathematical term below with its corresponding definition or workplace application.
A warehouse supervisor uses the formula 'Total Storage Cost = (Pallet Count) × (Daily Rate per Pallet)'. When the system reports a Total Storage Cost of $0, the supervisor applies the Zero Product Property to investigate the cause. Arrange the following steps in the correct logical order for applying this property's reasoning.
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In a logistics warehouse, a sorting machine is programmed with the formula (w - 15)(w - 30) = 0 to identify specific weight limits (w) for packages. To find the solutions, a technician sets each part of the formula to zero: w - 15 = 0 and w - 30 = 0. Which mathematical property justifies this specific step?
A software developer is writing a documentation comment for a function that calculates 'Projected Revenue' as the product of 'Market Share' and 'Total Market Size'. The developer needs to recall the Zero Product Property to explain what can be concluded when the Projected Revenue is $0. Which of the following is the correct formal definition of this property?
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Learn After
A quality control inspector at a manufacturing plant is checking the calibration of a machine part. The acceptable calibration settings, , are determined by solving the equation . Recalling the application of the Zero Product Property, what are the two required settings for ?
An HVAC technician is adjusting a pressure-limiting valve where the allowable pressure deviations (in psi) are determined by solving the equation . Arrange the following steps in the correct order to solve for based on the rule that if a product equals zero, at least one of its factors must be zero.
An industrial technician is solving the equation to calibrate a pressure-limiting valve. Match each stage of the solution process with the correct mathematical expression or calculation used in that step.
A logistics coordinator is using the equation to determine the equilibrium points for a delivery route. True or False: According to the Zero Product Property, the coordinator should solve this by setting each individual factor equal to zero, creating the equations and .
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