An electric device delivers a current of [tex]$15.0 A$[/tex] for 30 seconds. How many electrons flow through it?
Answer (1)
Expand the given equation: y = ( 2 x + 3 ) 2 = 4 x 2 + 12 x + 9 .
Identify the standard form: 0 = 4 x 2 + 12 x + 9 .
Determine the coefficients: a = 4 , b = 12 , c = 9 .
The values of a , b , c are a = 4 , b = 12 , c = 9 .
Explanation
Understanding the Problem We are given the quadratic equation y = ( 2 x + 3 ) 2 . Our goal is to express this equation in the standard form y = a x 2 + b x + c and then identify the values of the coefficients a , b , and c .
Expanding the Equation First, we need to expand the given equation. We have y = ( 2 x + 3 ) 2 = ( 2 x + 3 ) ( 2 x + 3 ) .
Performing the Multiplication Expanding the product, we get y = ( 2 x ) ( 2 x ) + ( 2 x ) ( 3 ) + ( 3 ) ( 2 x ) + ( 3 ) ( 3 ) = 4 x 2 + 6 x + 6 x + 9 = 4 x 2 + 12 x + 9 .
Identifying the Coefficients So the standard form of the quadratic equation is y = 4 x 2 + 12 x + 9 . Now we can identify the coefficients: a = 4 , b = 12 , and c = 9 .
Final Answer Therefore, the standard form of the given quadratic equation is 0 = 4 x 2 + 12 x + 9 , and the values of the coefficients are a = 4 , b = 12 , and c = 9 .
Examples
Understanding quadratic equations is crucial in various fields, such as physics and engineering. For instance, when calculating the trajectory of a projectile, the height of the projectile can be modeled by a quadratic equation. By expressing the equation in standard form, we can easily determine key parameters like the initial height, maximum height, and time of flight. This allows engineers to design systems and predict outcomes accurately.
