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Which expressions are equivalent to this expression?

[tex]\[30 x^2 - 5 x - 10\][/tex]

A. [tex]\[5 x(6 x - x - 2)\][/tex]

B. [tex]\[(10 x - 5)(3 x - 5)\][/tex]

C. [tex]\[5(2 x + 1)(3 x - 2)\][/tex]

D. [tex]\[-5(-6 x^2 + x + 2)\][/tex]

E. [tex]\[3 x(2 x - 1) + 2(2 x - 1)\][/tex]

F. [tex]\[5(2 x - 1)(3 x + 2)\][/tex]

Sagot :

GinnyAnsweridn
To determine which expressions are equivalent to [tex]\(30x^2 - 5x - 10\)[/tex], we need to perform algebraic manipulations to simplify each given expression and see if it matches the original polynomial. Here's the detailed, step-by-step solution:

### Step 1: Simplify each expression

1. Expression 1: [tex]\(5x(6x - x - 2)\)[/tex]
[tex]\[ 5x(6x - x - 2) = 5x(5x - 2) = 25x^2 - 10x \][/tex]

2. Expression 2: [tex]\((10x - 5)(3x - 5)\)[/tex]
[tex]\[ (10x - 5)(3x - 5) = 10x \cdot 3x - 10x \cdot 5 - 5 \cdot 3x + 5 \cdot 5 = 30x^2 - 50x - 15x + 25 = 30x^2 - 65x + 25 \][/tex]

3. Expression 3: [tex]\(5(2x + 1)(3x - 2)\)[/tex]
[tex]\[ 5(2x + 1)(3x - 2) = 5 \left(2x \cdot 3x - 2x \cdot 2 + 1 \cdot 3x - 1 \cdot 2\right) = 5 (6x^2 - 4x + 3x - 2)= 5(6x^2 - x - 2) = 30x^2 - 5x - 10 \][/tex]

4. Expression 4: [tex]\(-5(-6x^2 + x + 2)\)[/tex]
[tex]\[ -5(-6x^2 + x + 2) = -5(-6x^2) + (-5)(x) + (-5)(2) = 30x^2 - 5x - 10 \][/tex]

5. Expression 5: [tex]\(3x(2x - 1) + 2(2x - 1)\)[/tex]
[tex]\[ 3x(2x - 1) + 2(2x - 1) = 6x^2 - 3x + 4x - 2 = 6x^2 + x - 2 \][/tex]

6. Expression 6: [tex]\(5(2x - 1)(3x + 2)\)[/tex]
[tex]\[ 5(2x - 1)(3x + 2) = 5(2x \cdot 3x + 2x \cdot 2 - 1 \cdot 3x - 1 \cdot 2) = 5(6x^2 + 4x - 3x - 2) = 5(6x^2 + x - 2) = 30x^2 + 5x - 10 \][/tex]

### Step 2: Compare each simplified expression with the original

- Comparison Results:
- Expression 1: [tex]\(25x^2 - 10x \neq 30x^2 - 5x - 10\)[/tex]
- Expression 2: [tex]\(30x^2 - 65x + 25 \neq 30x^2 - 5x - 10\)[/tex]
- Expression 3: [tex]\(30x^2 - 5x - 10 = 30x^2 - 5x - 10\)[/tex]
- Expression 4: [tex]\(30x^2 - 5x - 10 = 30x^2 - 5x - 10\)[/tex]
- Expression 5: [tex]\(6x^2 + x - 2 \neq 30x^2 - 5x - 10\)[/tex]
- Expression 6: [tex]\(30x^2 + 5x - 10 \neq 30x^2 - 5x - 10\)[/tex]

### Step 3: Select the correct answers

Expressions 3 ([tex]\(5(2x+1)(3x-2)\)[/tex]) and 4 ([tex]\(-5(-6x^2 + x + 2)\)[/tex]) are equivalent to the given expression [tex]\(30x^2 - 5x - 10\)[/tex].

Thus, the correct answers are:
- [tex]\(5(2x+1)(3x-2)\)[/tex]
- [tex]\(-5(-6x^2 + x + 2)\)[/tex]
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