- Expand the binomial \((2x + 3)^2\) to get \(4x^2 + 12x + 9\).
- Set the expression equal to zero to arrive at the standard form \(0 = 4x^2 + 12x + 9\).
- Identify the coefficients by comparing with the general form \(ax^2 + bx + c = 0\).
- Final Answer: The correct option is \(0 = 4x^2 + 12x + 9\), with coefficients \(a = 4\), \(b = 12\), and \(c = 9\).
Explanation
- Problem Analysis
We are given the quadratic equation \(y = (2x + 3)^2\). Our objective is to rewrite this in the standard form \(ax^2 + bx + c = 0\) and identify the coefficients \(a, b,\) and \(c\). To do this, we need to expand the squared expression.
- Expanding the Expression
We expand the term \((2x + 3)^2\) using the binomial square identity \((A + B)^2 = A^2 + 2AB + B^2\). Here, \(A = 2x\) and \(B = 3\).
Substituting these into the identity:
\[(2x + 3)^2 = (2x)^2 + 2(2x)(3) + 3^2\]
\[(2x + 3)^2 = 4x^2 + 12x + 9\]
Thus, the expanded form is \(y = 4x^2 + 12x + 9\). Setting this equal to zero, we obtain the equation \(4x^2 + 12x + 9 = 0\).
- Identifying Coefficients and Final Selection
Comparing the expanded equation \(4x^2 + 12x + 9 = 0\) to the provided options, it matches the choice \(0 = 4x^2 + 12x + 9\). By matching our equation to the standard form \(ax^2 + bx + c = 0\), we identify the coefficients as:
\(a = 4\)
\(b = 12\)
\(c = 9\)
Examples
Quadratic equations like \(y = (2x + 3)^2\) appear frequently in physics and engineering, such as calculating the path of a projectile or determining the area of a square whose side length is variable. Understanding how to expand and write them in standard form is crucial for solving for roots or analyzing the behavior of functions in real-world modeling scenarios.