Which of the following equations has two distinct real roots ?

Options

A. `2x^(2) - 3 sqrt2 x + (9)/(4) = 0`
B. `x^(2) + x - 5 = 0`
C. `x^(2) + 3x + 2sqrt2 = 0`
D. `5x^(2) -3x+ 1 = 0`

Answer

To determine which of the given equations has two distinct real roots, we need to calculate the discriminant (D) for each equation. The discriminant is given by the formula:

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

where \( a \), \( b \), and \( c \) are the coefficients of the quadratic equation in the standard form \( ax^2 + bx + c = 0 \). An equation has two distinct real roots if \( D > 0 \).

Let’s analyze each option step by step.

### Step 1: Analyze the first equation
**Equation:** \( 2x^2 - 3\sqrt{2}x + \frac{9}{4} = 0 \)

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

\[ D = b^2 - 4ac = (-3\sqrt{2})^2 - 4(2)\left(\frac{9}{4}\right) \]

\[ D = 18 - 18 = 0 \]

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

### Step 2: Analyze the second equation
**Equation:** \( x^2 + x - 5 = 0 \)

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

\[ D = b^2 - 4ac = (1)^2 - 4(1)(-5) \]

\[ D = 1 + 20 = 21 \]

Since \( D = 21 > 0 \), this equation has two distinct real roots.

### Step 3: Analyze the third equation
**Equation:** \( x^2 - 3x + 2\sqrt{2} = 0 \)

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

\[ D = b^2 - 4ac = (-3)^2 - 4(1)(2\sqrt{2}) \]

\[ D = 9 - 8\sqrt{2} \]

Calculating \( 8\sqrt{2} \approx 11.31 \), we find:

\[ D \approx 9 - 11.31 \approx -2.31 \]

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

### Step 4: Analyze the fourth equation
**Equation:** \( 5x^2 - 3x + 1 = 0 \)

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

\[ D = b^2 - 4ac = (-3)^2 - 4(5)(1) \]

\[ D = 9 - 20 = -11 \]

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

### Conclusion
The only equation that has two distinct real roots is:

**Option 2: \( x^2 + x - 5 = 0 \)**

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