Question

Which TWO of the following expressions are equivalent to 20x^2 − 116x + 80?

Group of answer choices

(x − 5)(5x − 4)

4(x + 5)(5x + 4)

4(x − 5)(5x − 4)

2(10x − 58x + 40)

4(5x^2 − 29 + 80)


4x(5x − 29) + 20

Answer 1

To determine which two expressions are equivalent to \( 20x^2 - 116x + 80 \), let's first factor the original expression step by step.

1. **Factor out the GCD:**

The GCD of the coefficients \(20\), \(-116\), and \(80\) is \(4\).

\[
20x^2 - 116x + 80 = 4(5x^2 - 29x + 20)
\]

2. **Next, we will factor the quadratic inside the parentheses:**

We want to factor \(5x^2 - 29x + 20\).

We need two numbers that multiply to \(5 \cdot 20 = 100\) and add up to \(-29\). The numbers that work are \(-25\) and \(-4\).

Rewrite \( -29x\) as \(-25x - 4x\):

\[
5x^2 - 25x - 4x + 20
\]

Group the terms:

\[
(5x^2 - 25x) + (-4x + 20) = 5x(x - 5) - 4(x - 5)
\]

Factor out the common factor \( (x - 5)\):

\[
= (5x - 4)(x - 5)
\]

So, we combine this with our GCD factor:

\[
20x^2 - 116x + 80 = 4(5x - 4)(x - 5)
\]

3. **Now we compare with the provided options:**

- \((x - 5)(5x - 4)\) -> Matches, as we have \(4(5x - 4)(x - 5)\).
- \(4(x + 5)(5x + 4)\) -> Not equivalent.
- \(4(x - 5)(5x - 4)\) -> Matches exactly with our factorization \(4(5x - 4)(x - 5)\).
- \(2(10x - 58x + 40)\) -> Not equivalent.
- \(4(5x^2 - 29 + 80)\) -> Not equivalent.
- \(4x(5x - 29) + 20\) -> Not equivalent.

The two equivalent expressions to \(20x^2 - 116x + 80\) are:

1. \((x - 5)(5x - 4)\)
2. \(4(x - 5)(5x - 4)\)

Thus, the correct answers are:

- (x − 5)(5x − 4)
- 4(x − 5)(5x − 4)