Question

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

Answer 1

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:

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