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:
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Factor out the greatest common factor (GCF):
The GCF of \( 4x^2, -20x, \) and \( -24 \) is \( 4 \). So we can factor it out:
\[ 4(x^2 - 5x - 6) \]
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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:
\[ x^2 - 5x - 6 = (x - 6)(x + 1) \]
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Putting it all together, we have:
\[ 4(x - 6)(x + 1) \]
Now, we can generate equivalent expressions that yield the same trinomial:
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First Expression: \( (2x - 12)(2x + 2) \) – When expanded, it simplifies to \( 4x^2 - 20x - 24 \).
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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!
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