Two resistors connected in series have an equivalent...

Two resistors connected in series have an equivalent resistance of $690 \Omega .$ When they are connected in parallel, their equivalent resistance is $150 \Omega .$ Find the resistance of each resistor.

Key Concepts

Series Connection

In a series circuit, resistors are connected end-to-end, and their resistances add together to form the equivalent resistance. This means that the overall resistance is the sum of the individual resistances, a concept crucial for analyzing circuits with components arranged in a series configuration.

Parallel Connection

In a parallel circuit, resistors are connected so that each resistor has the same voltage across it. The equivalent resistance is found by taking the reciprocal of the sum of the reciprocals of the individual resistances, or, for two resistors, using the product-over-sum formula. This method is fundamental for reducing complex resistor networks into simpler forms.

System of Equations

When dealing with circuits that involve both series and parallel arrangements, it becomes necessary to set up and solve a system of equations. Each equation is derived from the respective formulas for equivalent resistance in series and parallel configurations, allowing for the determination of individual resistor values.

Quadratic Equations

The process of solving the system of equations from the series and parallel resistance formulas can lead to a quadratic equation. Solving quadratic equations is an essential mathematical tool in circuit analysis, enabling the extraction of individual component values from the derived relationships.

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