Which of the following equations has no real roots?

Options

A. `x^(2)-4x+3sqrt(2)=0`
B. `x^(2)+4x-3sqrt(2)=0`
C. `x^(2)-4x-3sqrt(2)=0`
D. `3x^(2)+4sqrt(3)x+4=0`

Explanation

To determine which of the given equations has no real roots, we need to use the concept of the discriminant from the quadratic formula. The discriminant (D) is given by the formula:

\[ D = b^2 - 4ac \]

A quadratic equation has no real roots if the discriminant is less than zero (D < 0). Let's analyze each option step by step.

### Step 1: Analyze the first option
For the first option, we have:
- \( a = 1 \)
- \( b = -4 \)
- \( c = 3\sqrt{2} \)

Calculate the discriminant:
\[ D = (-4)^2 - 4 \cdot 1 \cdot 3\sqrt{2} \]
\[ D = 16 - 12\sqrt{2} \]

Since \( 12\sqrt{2} \) is approximately \( 16.97 \), we have:
\[ D = 16 - 12\sqrt{2} < 0 \]

Thus, the first option has no real roots.

### Step 2: Analyze the second option
For the second option, we have:
- \( a = 1 \)
- \( b = 4 \)
- \( c = -3\sqrt{2} \)

Calculate the discriminant:
\[ D = 4^2 - 4 \cdot 1 \cdot (-3\sqrt{2}) \]
\[ D = 16 + 12\sqrt{2} \]

Since \( 12\sqrt{2} \) is positive, we have:
\[ D = 16 + 12\sqrt{2} > 0 \]

Thus, the second option has real roots.

### Step 3: Analyze the third option
For the third option, we have:
- \( a = 1 \)
- \( b = -4 \)
- \( c = -3\sqrt{2} \)

Calculate the discriminant:
\[ D = (-4)^2 - 4 \cdot 1 \cdot (-3\sqrt{2}) \]
\[ D = 16 + 12\sqrt{2} \]

Again, since \( 12\sqrt{2} \) is positive, we have:
\[ D = 16 + 12\sqrt{2} > 0 \]

Thus, the third option also has real roots.

### Step 4: Analyze the fourth option
For the fourth option, we have:
- \( a = 3 \)
- \( b = 4\sqrt{3} \)
- \( c = 4 \)

Calculate the discriminant:
\[ D = (4\sqrt{3})^2 - 4 \cdot 3 \cdot 4 \]
\[ D = 48 - 48 = 0 \]

Since the discriminant is zero, the fourth option has one real root.

### Conclusion
The only option that has no real roots is the **first option**.