An electric device delivers a current of [tex]$15.0 A$[/tex] for 30 seconds. How many electrons flow through it?
Answer (2)
Factor the quadratic expression x 2 + x − 72 .
Find two numbers a and b such that a × b = − 72 and a + b = 1 .
The numbers are 9 and − 8 , since 9 × − 8 = − 72 and 9 + ( − 8 ) = 1 .
Therefore, x 2 + x − 72 = ( x + 9 ) ( x − 8 ) , and the answer is ( x + 9 ) ( x − 8 ) .
Explanation
Understanding the Problem We are given the quadratic expression x 2 + x − 72 and asked to rewrite it in the form ( x + a ) ( x + b ) . This requires us to factor the quadratic expression.
Finding the Factors To factor the quadratic expression x 2 + x − 72 , we need to find two numbers a and b such that their product is equal to the constant term − 72 and their sum is equal to the coefficient of the linear term, which is 1 .
Identifying the Correct Numbers We are looking for two numbers a and b such that a × b = − 72 and a + b = 1 . By trial and error or by considering factor pairs of − 72 , we can find that the numbers are 9 and − 8 , since 9 × − 8 = − 72 and 9 + ( − 8 ) = 1 .
Writing the Factored Form Therefore, we can write the quadratic expression as ( x + 9 ) ( x − 8 ) . So, a = 9 and b = − 8 (or vice versa).
Final Answer The factored form of the given expression is ( x + 9 ) ( x − 8 ) .
Examples
Factoring quadratic expressions is a fundamental skill in algebra and is used in many real-world applications. For example, engineers use factoring to design structures, ensuring stability and optimal use of materials. Imagine designing a rectangular garden where you know the area and need to find the dimensions. If the area is represented by a quadratic expression, factoring helps determine the possible lengths and widths of the garden. This skill is also crucial in physics for solving equations related to motion and forces.
About 2.81 × 1 0 21 electrons flow through the device when it operates with a current of 15.0 A for 30 seconds. This is calculated by first determining the total charge (450 Coulombs) and then dividing that by the charge of a single electron (1.602 x 10^-19 Coulombs). Hence, the total number of electrons flowing is significant, illustrating the large quantities of charge moving in electrical circuits.
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