Jimmy is attempting to factor trinomials. Which expressions are equivalent to 32b2 − 80b + 50?

Select TWO correct answers.

Group of answer choices

2(4b – 5)(4b ‒ 5)

(4b – 5)(4b ‒ 5)

2(4b – 5)2

-2(4b – 5)2

1 answer

To factor the expression \(32b^2 - 80b + 50\), we can first look for a common factor. The greatest common factor of the coefficients (32, -80, 50) is 2.

So we can factor out 2:

\[ 32b^2 - 80b + 50 = 2(16b^2 - 40b + 25) \]

Next, we can factor the quadratic \(16b^2 - 40b + 25\). This is a perfect square trinomial and can be factored as:

\[ 16b^2 - 40b + 25 = (4b - 5)^2 \]

So, combining these, we have:

\[ 32b^2 - 80b + 50 = 2(4b - 5)^2 \]

Now, let’s analyze the provided options:

  1. \(2(4b - 5)(4b - 5)\): This is equivalent to \(2(4b - 5)^2\). This is correct.

  2. \((4b - 5)(4b - 5)\): This is simply \((4b - 5)^2\) and does not have the factor of 2. This is not correct.

  3. \(2(4b - 5)^2\): This matches our derived factorization exactly. This is correct.

  4. \(-2(4b - 5)^2\): This has the opposite sign of what we factored; hence, it is not equivalent. This is not correct.

The correct choices are:

  • \(2(4b - 5)(4b - 5)\)
  • \(2(4b - 5)^2\)