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 to get y = 4 x 2 + 12 x + 9 .
Rewrite the equation in the standard form by setting it to zero: 0 = 4 x 2 + 12 x + 9 − y .
Consider the case where y = 0 to find the standard form of the given equation: 0 = 4 x 2 + 12 x + 9 .
Compare the result with the given options and choose the correct one: 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 . To find the standard form, we need to expand the given equation and rearrange it to equal zero.
Expanding the Equation First, let's expand the given equation: y = ( 2 x + 3 ) 2
y = ( 2 x + 3 ) ( 2 x + 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 express this equation in the standard form, which means setting it equal to zero. To do this, we subtract y from both sides: 0 = 4 x 2 + 12 x + 9 − y
Setting y=0 Since we are looking for the standard form of the given quadratic equation, we consider the case where y = 0 . This gives us: 0 = 4 x 2 + 12 x + 9
Choosing the Correct Option Now we compare this result with the given options:
0 = 4 x 2 + 10 x + 6
0 = 25 x 2
0 = 4 x 2 + 9
0 = 4 x 2 + 12 x + 9
The correct option is 0 = 4 x 2 + 12 x + 9 .
Examples
Quadratic equations are used in various real-life applications, such as modeling the trajectory of a projectile, designing parabolic mirrors and reflectors, and determining the minimum or maximum values in optimization problems. For example, if you want to build a parabolic bridge, you need to solve a quadratic equation to determine the shape and dimensions of the bridge. Similarly, if you are launching a rocket, you need to understand the quadratic relationship between the launch angle and the range of the rocket to maximize the distance it travels.
