Which of the following equations has real roots ?

Options

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

Answer

To determine which of the given equations has real roots, we will use the discriminant method. The discriminant \( D \) of a quadratic equation in the form \( Ax^2 + Bx + C = 0 \) is given by the formula:

\[ D = B^2 - 4AC \]

If \( D \geq 0 \), the equation has real roots. If \( D < 0 \), the equation has imaginary roots.

Let's analyze each option step by step:

### Option 1: \( 3x^2 + 4x + 5 = 0 \)

1. Identify coefficients: \( A = 3 \), \( B = 4 \), \( C = 5 \).
2. Calculate the discriminant:
\[ D = B^2 - 4AC = 4^2 - 4 \times 3 \times 5 = 16 - 60 = -44 \]
3. Since \( D < 0 \), this equation does not have real roots.

### Option 2: \( x^2 + x + 4 = 0 \)

1. Identify coefficients: \( A = 1 \), \( B = 1 \), \( C = 4 \).
2. Calculate the discriminant:
\[ D = B^2 - 4AC = 1^2 - 4 \times 1 \times 4 = 1 - 16 = -15 \]
3. Since \( D < 0 \), this equation does not have real roots.

### Option 3: \( (x - 1)(2x - 5) = 0 \)

1. This equation is already factored. The roots can be found by setting each factor to zero:
- \( x - 1 = 0 \) gives \( x = 1 \).
- \( 2x - 5 = 0 \) gives \( 2x = 5 \) or \( x = \frac{5}{2} \).
2. Both roots \( x = 1 \) and \( x = \frac{5}{2} \) are real numbers.

### Option 4: \( 2x^2 - 3x + 4 = 0 \)

1. Identify coefficients: \( A = 2 \), \( B = -3 \), \( C = 4 \).
2. Calculate the discriminant:
\[ D = B^2 - 4AC = (-3)^2 - 4 \times 2 \times 4 = 9 - 32 = -23 \]
3. Since \( D < 0 \), this equation does not have real roots.

### Conclusion

After checking all options, only **Option 3** has real roots.