Given $y=(2 x+3)^2$, choose the standard form of the given quadratic equation.
The given equation is \(y = (2x + 3)^2\). A quadratic equation in standard form is generally written as \(ax^2 + bx + c = 0\), where \(a, b,\) and \(c\) are constants and \(a eq 0\). Our goal is to expand the given expression \((2x + 3)^2\) to identify the equivalent standard quadratic form from the provided options.
To expand the expression \((2x + 3)^2\), we use the algebraic identity for the square of a binomial: \((A + B)^2 = A^2 + 2AB + B^2\). Here, we let \(A = 2x\) and \(B = 3\). Substituting these values into the identity:
\[(2x + 3)^2 = (2x)^2 + 2(2x)(3) + (3)^2\]
Calculating each term:
\((2x)^2 = 4x^2\)
\(2(2x)(3) = 12x\)
\(3^2 = 9\)
Combining these results, we get the expanded form: \(4x^2 + 12x + 9\).
Now that we have expanded the expression as \(4x^2 + 12x + 9\), we can set the quadratic equation equal to zero, which is the standard procedure for finding the root-related form or the equation defined by the polynomial:
\[0 = 4x^2 + 12x + 9\]
Comparing this result to the given options:
We see that option (c) matches our derived expression perfectly.
Quadratic equations are fundamental in modeling real-world physical phenomena. For example, if you throw a ball, its height over time is often described by a quadratic equation, such as \(h(t) = -gt^2 + v_0t + h_0\). Understanding how to expand and manipulate these expressions allows scientists and engineers to determine key characteristics, such as when the object hits the ground (by setting the expression to zero), how long it stays in the air, or its maximum height. Mastering these algebraic techniques provides a solid foundation for physics, engineering, and data analysis.