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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

1 answer

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

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