Which of the following equations has real roots?

Options

A. `2x^(2)-3x+4=0`
B. `(x-1)(2x-5)=0`
C. `3x^(2)+4x+5=0`
D. Cannot be determined

Answer

To determine which of the given equations has real roots, we need to analyze the discriminant of each quadratic equation. The discriminant (D) is calculated using the formula:

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

Where \( a \), \( b \), and \( c \) are the coefficients of the quadratic equation in the form \( ax^2 + bx + c = 0 \). The roots of the quadratic equation are real if \( D \geq 0 \).

### Step-by-Step Solution:

1. **First Equation: \( 2x^2 - 3x - 4 = 0 \)**
- Here, \( a = 2 \), \( b = -3 \), and \( c = -4 \).
- Calculate the discriminant:
\[ D = (-3)^2 - 4 \cdot 2 \cdot (-4) = 9 + 32 = 41 \]
- Since \( D > 0 \), this equation has real roots.

2. **Second Equation: \( (x - 1)(2x - 5) = 0 \)**
- Expand the equation:
\[ 2x^2 - 5x - 2x + 5 = 2x^2 - 7x + 5 = 0 \]
- Here, \( a = 2 \), \( b = -7 \), and \( c = 5 \).
- Calculate the discriminant:
\[ D = (-7)^2 - 4 \cdot 2 \cdot 5 = 49 - 40 = 9 \]
- Since \( D > 0 \), this equation also has real roots.

3. **Third Equation: \( 3x^2 - 4x + 5 = 0 \)**
- Here, \( a = 3 \), \( b = -4 \), and \( c = 5 \).
- Calculate the discriminant:
\[ D = (-4)^2 - 4 \cdot 3 \cdot 5 = 16 - 60 = -44 \]
- Since \( D < 0 \), this equation does not have real roots.

### Conclusion:
The first two equations \( 2x^2 - 3x - 4 = 0 \) and \( (x - 1)(2x - 5) = 0 \) have real roots, while the third equation \( 3x^2 - 4x + 5 = 0 \) does not.