Question

When simplified, which expressions produce the trinomial 4x^2 - 20x - 24 pick two that apply

Answer 1

To find expressions that simplify to the trinomial \( 4x^2 - 20x - 24 \), we can factor the trinomial or manipulate expressions algebraically.

First, let's factor the trinomial:

1. Factor out the greatest common factor (GCF):

The GCF of \( 4x^2, -20x, \) and \( -24 \) is \( 4 \). So we can factor it out:

\[
4(x^2 - 5x - 6)
\]

2. Now, we can factor the quadratic expression \( x^2 - 5x - 6 \):

We need two numbers that multiply to \( -6 \) and add up to \( -5 \). These numbers are \( -6 \) and \( 1 \).

Therefore, we can factor:

\[
x^2 - 5x - 6 = (x - 6)(x + 1)
\]

3. Putting it all together, we have:

\[
4(x - 6)(x + 1)
\]

Now, we can generate equivalent expressions that yield the same trinomial:

1. **First Expression**: \( (2x - 12)(2x + 2) \) – When expanded, it simplifies to \( 4x^2 - 20x - 24 \).

2. **Second Expression**: \( 4(x - 6)(x + 1) \) – This is our factored form, and clearly simplifies directly back to \( 4x^2 - 20x - 24 \).

Thus, any expressions like \( 2(2x^2 - 10x - 12) \) would also yield \( 4x^2 - 20x - 24 \) as well because it's a scalar multiplication of the same equation.

So, the two expressions that apply can indeed be:

- \( (2x - 12)(2x + 2) \)
- \( 4(x - 6)(x + 1) \)

Other valid expressions might include variations like \( 4x(x - 5) - 4 \times 6 \) or other scalars that retain the equality to the original trinomial when simplified!