Question

Which expressions are equivalent to 6x2+36x+54
?

Select TWO correct answers.
A.6(x−3)2

B.6(x+3)2

C.3(x2−6x+9)

D.6(x2+6x+9)

E.3(x−9)2

Answer 1

To determine which expressions are equivalent to \(6x^2 + 36x + 54\), we can factor the original expression and compare it to the options provided.

1. **Starting with the expression:**
\[
6x^2 + 36x + 54
\]

2. **Factoring out the greatest common factor (GCF):**
\[
6(x^2 + 6x + 9)
\]

3. **Recognizing that \(x^2 + 6x + 9\) is a perfect square:**
\[
x^2 + 6x + 9 = (x + 3)^2
\]
Thus,
\[
6(x^2 + 6x + 9) = 6(x + 3)^2
\]

Now we have:
\[
6x^2 + 36x + 54 = 6(x + 3)^2
\]

4. **Reviewing the provided options:**

- **A. \(6(x−3)^2\)**: Not equivalent (since it has \(x-3\) instead of \(x+3\)).
- **B. \(6(x+3)^2\)**: Equivalent (matches the factored form we derived).
- **C. \(3(x^2−6x+9)\)**: Not equivalent (since \(x^2 - 6x + 9 = (x-3)^2\)).
- **D. \(6(x^2+6x+9)\)**: Equivalent (as this is just the rearrangement of the expression we derived).
- **E. \(3(x−9)^2\)**: Not equivalent (since it has \(x-9\) instead of \(x+3\)).

5. **Final Choices:**
The two correct answers that are equivalent to \(6x^2 + 36x + 54\) are:
- **B. \(6(x+3)^2\)**
- **D. \(6(x^2+6x+9)\)**