Question

use the dropdown menu to indicate whether each expression below is equivalent to $4x^2 - 9$.

a. $(2x - 3)^2 + 12x + 9$

b. $(2x - 3)(2x + 3)$

c. $(2x + 3)^2 - 12x - 18$

d. $(4x + 9)(x - 1)$

Explanation:

Part a

Step1: Expand \((2x - 3)^2\)

Using the formula \((a - b)^2=a^{2}-2ab + b^{2}\), where \(a = 2x\) and \(b=3\), we get \((2x)^{2}-2\times(2x)\times3+3^{2}=4x^{2}-12x + 9\)

Step2: Add \(12x + 9\) to the expanded form

\((4x^{2}-12x + 9)+(12x + 9)=4x^{2}-12x + 9+12x + 9=4x^{2}+18\)
Since \(4x^{2}+18
eq4x^{2}-9\), the expression is not equivalent.

Part b

Step1: Use the difference of squares formula \((a - b)(a + b)=a^{2}-b^{2}\)

Here \(a = 2x\) and \(b = 3\), so \((2x-3)(2x + 3)=(2x)^{2}-3^{2}=4x^{2}-9\)
Since the result is equal to \(4x^{2}-9\), the expression is equivalent.

Part c

Step1: Expand \((2x + 3)^2\)

Using the formula \((a + b)^2=a^{2}+2ab + b^{2}\), where \(a=2x\) and \(b = 3\), we get \((2x)^{2}+2\times(2x)\times3+3^{2}=4x^{2}+12x + 9\)

Step2: Subtract \(12x+18\) from the expanded form

\((4x^{2}+12x + 9)-(12x + 18)=4x^{2}+12x + 9-12x - 18=4x^{2}-9\)
Since the result is equal to \(4x^{2}-9\), the expression is equivalent.

Part d

Answer:

a. Not Equivalent
b. Equivalent
c. Equivalent
d. Not Equivalent