Find the values of $a, b$, and $c$. Given $y=(2 x+3)^2$, choose the standard form of the given quadratic equation.
A. $0=25 x^2$
B. $0=4 x^2+9$
C. $0=4 x^2+10 x+6$
D. $0=4 x^2+12 x+9
Answer (2)
Expand the given equation: ( 2 x + 3 ) 2 = 4 x 2 + 12 x + 9 .
Rewrite the equation in standard form: 0 = 4 x 2 + 12 x + 9 .
Compare the expanded form with the given options.
The correct standard form is 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 To find the standard form, we need to expand the given equation. Expanding ( 2 x + 3 ) 2 means multiplying ( 2 x + 3 ) by itself: ( 2 x + 3 ) 2 = ( 2 x + 3 ) ( 2 x + 3 ) Using the distributive property (also known as the FOIL method), we get: ( 2 x + 3 ) ( 2 x + 3 ) = ( 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 So, y = 4 x 2 + 12 x + 9 .
Rewriting in Standard Form Now, we need to rewrite the equation in the standard form, which is a x 2 + b x + c = 0 . To do this, we set y to zero: 0 = 4 x 2 + 12 x + 9 This matches one of the given options.
Choosing the Correct Option Comparing our expanded form 0 = 4 x 2 + 12 x + 9 with the given options:
0 = 25 x 2
0 = 4 x 2 + 9
0 = 4 x 2 + 10 x + 6
0 = 4 x 2 + 12 x + 9
We see that the correct option is 0 = 4 x 2 + 12 x + 9 .
Examples
Understanding quadratic equations is crucial in various real-world applications. For instance, engineers use them to design parabolic mirrors and suspension bridges. Architects apply quadratic equations to calculate the curves in arches and domes, ensuring structural integrity and aesthetic appeal. Moreover, in physics, these equations help model projectile motion, enabling accurate predictions of trajectories for objects like balls or rockets.
The standard form of the quadratic equation given by y = ( 2 x + 3 ) 2 is 0 = 4 x 2 + 12 x + 9 . This matches option D from the choices provided. Therefore, the correct answer is D.
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