Given $y=(2 x+3)^2$, choose the standard form of the given quadratic equation.
A. $0=4 x^2+10 x+6$
B. $0=25 x^2$
C. $0=4 x^2+9$
D. $0=4 x^2+12 x+9$
Answer (1)
Expand the given equation: y = ( 2 x + 3 ) 2 becomes y = 4 x 2 + 12 x + 9 .
Rewrite the expanded equation in the standard form: 0 = 4 x 2 + 12 x + 9 − y .
Compare the derived standard form with the given options.
Choose the correct option: 0 = 4 x 2 + 12 x + 9 .
Explanation
Understanding the Problem We are given the quadratic equation y = ( 2 x + 3 ) 2 and asked to choose its standard form from the given options. The standard form of a quadratic equation is a x 2 + b x + c = 0 .
Expanding the Equation First, we need to expand the given equation:
y = ( 2 x + 3 ) 2 y = ( 2 x + 3 ) ( 2 x + 3 ) y = ( 2 x ) ( 2 x ) + ( 2 x ) ( 3 ) + ( 3 ) ( 2 x ) + ( 3 ) ( 3 ) y = 4 x 2 + 6 x + 6 x + 9 y = 4 x 2 + 12 x + 9
Rewriting in Standard Form Now, we want to rewrite the expanded equation in the standard form a x 2 + b x + c = 0 . Since we are given options in the form of 0 = ... , we can rewrite our equation as:
0 = 4 x 2 + 12 x + 9 − y
However, we are looking for an equation where y = 0 . So, we need to consider the case where y = 0 is not explicitly stated, but implied by the form of the answer choices. The correct standard form should match the coefficients we found after expanding.
Comparing with Options Comparing our expanded form 4 x 2 + 12 x + 9 with the given options:
0 = 4 x 2 + 10 x + 6 (Incorrect)
0 = 25 x 2 (Incorrect)
0 = 4 x 2 + 9 (Incorrect)
0 = 4 x 2 + 12 x + 9 (Correct)
Final Answer Therefore, the correct standard form of the given quadratic equation is 0 = 4 x 2 + 12 x + 9 .
Examples
Quadratic equations are used in various real-life applications, such as calculating the trajectory of a ball, designing parabolic mirrors, or determining the optimal dimensions of a rectangular garden to maximize its area. Understanding how to convert a quadratic equation into its standard form is essential for solving these types of problems efficiently. For example, if you want to find the roots of the equation (where the trajectory hits the ground), having it in standard form allows you to easily apply the quadratic formula or factoring techniques.
