Simplify the expression `2/t-3-2t` and find the values of t for which the expression is 0.

Options

A. `-2,-1`
B. `-2,1/2`
C. `-2,-1/2`
D. `2,1`

Answer

To simplify the expression \( \frac{2}{t} - 3 - 2t \) and find the values of \( t \) for which the expression equals 0, we will follow these steps:

### Step 1: Rewrite the Expression
We start with the expression:
\[ \frac{2}{t} - 3 - 2t \]

### Step 2: Find a Common Denominator
The common denominator for the terms in the expression is \( t \). We can rewrite the expression as:
\[ \frac{2}{t} - \frac{3t}{t} - \frac{2t^2}{t} \]
This gives us:
\[ \frac{2 - 3t - 2t^2}{t} \]

### Step 3: Rearrange the Numerator
Now, we can rearrange the numerator:
\[ \frac{-2t^2 - 3t + 2}{t} \]

### Step 4: Set the Expression Equal to Zero
To find the values of \( t \) for which the expression equals 0, we set the numerator equal to zero:
\[ -2t^2 - 3t + 2 = 0 \]

### Step 5: Multiply by -1
To simplify the equation, we can multiply through by -1:
\[ 2t^2 + 3t - 2 = 0 \]

### Step 6: Factor the Quadratic Equation
Next, we need to factor the quadratic equation. We are looking for two numbers that multiply to \( 2 \times -2 = -4 \) and add up to \( 3 \). The numbers \( 4 \) and \( -1 \) work because:
\[ 4 \cdot (-1) = -4 \quad \text{and} \quad 4 + (-1) = 3 \]

### Step 7: Rewrite the Equation
We can rewrite the quadratic as:
\[ 2t^2 + 4t - t - 2 = 0 \]

### Step 8: Group the Terms
Now, we group the terms:
\[ (2t^2 + 4t) + (-t - 2) = 0 \]

### Step 9: Factor by Grouping
Factoring by grouping gives us:
\[ 2t(t + 2) - 1(t + 2) = 0 \]
This can be factored further as:
\[ (2t - 1)(t + 2) = 0 \]

### Step 10: Solve for \( t \)
Now we can set each factor equal to zero:
1. \( 2t - 1 = 0 \) leads to \( t = \frac{1}{2} \)
2. \( t + 2 = 0 \) leads to \( t = -2 \)

### Final Values of \( t \)
Thus, the values of \( t \) for which the expression is equal to 0 are:
\[ t = \frac{1}{2} \quad \text{and} \quad t = -2 \]

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