Identify two expressions that will be equivalent to 2x^2−8x−10

Question

when distributed. Enter the option with the lesser number first.

Option #1: 2x(x−4x−5)

Option #2: (2x+2)(x−5)

Option #3: 2x(x−5)+2(x−5)

Option #4: (x−5)(x−3)

Option #5: −6x−10

(1 point)
Option # ( )
and Option # ( )

Answer 1

To determine which expressions are equivalent to \( 2x^2 - 8x - 10 \), let's simplify each option and see which ends up being equivalent to the original expression.

1. Option #1: \( 2x(x - 4x - 5) \)

   • This appears to be a miswritten expression. It should be simplified correctly. Assuming it is \( 2x(x - 4) - 5 \):
   • Rewrite: \( 2x(-3x - 5) \)
   • When distributed: \( -6x^2 - 10x \) (not equivalent)

2. Option #2: \( (2x + 2)(x - 5) \)

   • Distributing: \[ 2x \cdot x + 2 \cdot x - 2x \cdot 5 - 2 \cdot 5 = 2x^2 + 2x - 10x - 10 = 2x^2 - 8x - 10 \]
   • This one is equivalent.

3. Option #3: \( 2x(x - 5) + 2(x - 5) \)

   • Factor out \( (x - 5) \): \[ (2x + 2)(x - 5) = 2x^2 - 8x - 10 \text{ (seen in Option 2)} \]
   • When combining: \[ 2x^2 - 10x + 2x - 10 = 2x^2 - 8x - 10 \]
   • This one is also equivalent.

4. Option #4: \( (x - 5)(x - 3) \)

   • Distributing: \[ x^2 - 3x - 5x + 15 = x^2 - 8x + 15 \text{ (not equivalent)} \]

5. Option #5: \( -6x - 10 \)

   • This is just a linear expression and cannot satisfy the quadratic equation.

Now, we find that Options #2 and #3 are equivalent to \( 2x^2 - 8x - 10 \).

So, the options with the lesser number first are:

Option # (2) and Option # (3)