Which of the following equations has two distinct real roots?

Options

A. `2x^(2)-3sqrt(2)x+9/4=0`
B. `x^(2)+x-5=0`
C. `x^(2)+3x+2sqrt(2)=0`
D. `5x^(2)-3x+1=0`

Explanation

To determine which of the given quadratic equations has two distinct real roots, we need to use the concept of the discriminant. The discriminant \( D \) of a quadratic equation in the form \( ax^2 + bx + c = 0 \) is given by the formula:

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

A quadratic equation has:
- Two distinct real roots if \( D > 0 \)
- One real root (or two equal roots) if \( D = 0 \)
- No real roots if \( D < 0 \)

Now, let's analyze each option step by step.

### Step 1: Analyze the first equation

Assume the first equation is \( 2x^2 - 3\sqrt{2}x + \frac{9}{4} = 0 \).

- Here, \( a = 2 \), \( b = -3\sqrt{2} \), and \( c = \frac{9}{4} \).
- Calculate the discriminant:

\[ D = (-3\sqrt{2})^2 - 4 \cdot 2 \cdot \frac{9}{4} \]
\[ D = 18 - 18 = 0 \]

Since \( D = 0 \), this equation does not have two distinct real roots.

### Step 2: Analyze the second equation

Assume the second equation is \( x^2 + x - 5 = 0 \).

- Here, \( a = 1 \), \( b = 1 \), and \( c = -5 \).
- Calculate the discriminant:

\[ D = (1)^2 - 4 \cdot 1 \cdot (-5) \]
\[ D = 1 + 20 = 21 \]

Since \( D > 0 \), this equation has two distinct real roots.

### Step 3: Analyze the third equation

Assume the third equation is \( x^2 + 3x + 2\sqrt{2} = 0 \).

- Here, \( a = 1 \), \( b = 3 \), and \( c = 2\sqrt{2} \).
- Calculate the discriminant:

\[ D = (3)^2 - 4 \cdot 1 \cdot (2\sqrt{2}) \]
\[ D = 9 - 8\sqrt{2} \]

Since \( \sqrt{2} \approx 1.414 \), we have \( 8\sqrt{2} \approx 11.312 \), thus:

\[ D \approx 9 - 11.312 < 0 \]

Since \( D < 0 \), this equation does not have two distinct real roots.

### Step 4: Analyze the fourth equation

Assume the fourth equation is \( 5x^2 - 3x + 1 = 0 \).

- Here, \( a = 5 \), \( b = -3 \), and \( c = 1 \).
- Calculate the discriminant:

\[ D = (-3)^2 - 4 \cdot 5 \cdot 1 \]
\[ D = 9 - 20 = -11 \]

Since \( D < 0 \), this equation does not have two distinct real roots.

### Conclusion

Among the given options, only the second equation \( x^2 + x - 5 = 0 \) has two distinct real roots.