`2x^2 -4x=t`
In the equation above, t is a constant. If the equation has no real solutions, which of the following could be the value of t ?

Options

A. `-3`
B. `-1`
C. `1`
D. 3

Explanation

To determine the value of \( t \) for which the equation \( 2x^2 - 4x = t \) has no real solutions, we need to analyze the quadratic equation formed by rearranging it into standard form.

### Step-by-step Solution:

1. **Rearranging the equation**:
Start with the given equation:
\[ 2x^2 - 4x - t = 0 \]
This is now in the standard quadratic form \( ax^2 + bx + c = 0 \), where:
- \( a = 2 \)
- \( b = -4 \)
- \( c = -t \)

**Hint**: Identify the coefficients \( a \), \( b \), and \( c \) from the standard form of a quadratic equation.

2. **Using the discriminant**:
For a quadratic equation to have no real solutions, the discriminant must be less than zero. The discriminant \( D \) is given by:
\[ D = b^2 - 4ac \]
Substituting the values of \( a \), \( b \), and \( c \):
\[ D = (-4)^2 - 4 \cdot 2 \cdot (-t) \]

**Hint**: Remember that the discriminant determines the nature of the roots of the quadratic equation.

3. **Calculating the discriminant**:
Calculate \( D \):
\[ D = 16 + 8t \]
We want this discriminant to be less than zero for the equation to have no real solutions:
\[ 16 + 8t < 0 \]

**Hint**: Set up the inequality to find the conditions on \( t \).

4. **Solving the inequality**:
Rearranging the inequality gives:
\[ 8t < -16 \]
Dividing both sides by 8:
\[ t < -2 \]

**Hint**: Isolate \( t \) to find the range of values that satisfy the condition.

5. **Finding possible values of \( t \)**:
Now, we need to find a value of \( t \) that is less than -2. For example, \( t = -3 \) satisfies this condition.

**Hint**: Check the options provided to see which values are less than -2.

### Conclusion:
The value of \( t \) that could make the equation \( 2x^2 - 4x = t \) have no real solutions is \( t = -3 \).