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 following equations has 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 from the quadratic equation \( ax^2 + bx + c = 0 \). If \( D > 0 \), the equation has two distinct real roots; if \( D = 0 \), it has exactly one real root; and if \( D < 0 \), it has no real roots.

Let's analyze each of the provided equations step by step.

### Step 1: Analyze the first equation \( 3x^2 + 4x + 5 = 0 \)

1. Identify coefficients:
- \( a = 3 \)
- \( b = 4 \)
- \( c = 5 \)

2. Calculate the discriminant:
\[ D = b^2 - 4ac = 4^2 - 4 \cdot 3 \cdot 5 = 16 - 60 = -44 \]
3. Since \( D < 0 \), this equation does not have real roots.

### Step 2: Analyze the second equation \( x^2 + x + 4 = 0 \)

1. Identify coefficients:
- \( a = 1 \)
- \( b = 1 \)
- \( c = 4 \)

2. Calculate the discriminant:
\[ D = b^2 - 4ac = 1^2 - 4 \cdot 1 \cdot 4 = 1 - 16 = -15 \]
3. Since \( D < 0 \), this equation does not have real roots.

### Step 3: Analyze the third equation \( x - 1(2x - 5) = 0 \)

1. Expand the equation:
\[ x - 1(2x - 5) = 0 \implies 2x^2 - 5x - 2x + 5 = 0 \implies 2x^2 - 7x + 5 = 0 \]

2. Identify coefficients:
- \( a = 2 \)
- \( b = -7 \)
- \( c = 5 \)

3. Calculate the discriminant:
\[ D = b^2 - 4ac = (-7)^2 - 4 \cdot 2 \cdot 5 = 49 - 40 = 9 \]
4. Since \( D > 0 \), this equation has two distinct real roots.

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

1. Identify coefficients:
- \( a = 2 \)
- \( b = -3 \)
- \( c = 4 \)

2. Calculate the discriminant:
\[ D = b^2 - 4ac = (-3)^2 - 4 \cdot 2 \cdot 4 = 9 - 32 = -23 \]
3. Since \( D < 0 \), this equation does not have real roots.

### Conclusion

The only equation that has real roots is the third equation \( 2x^2 - 7x + 5 = 0 \).

### Final Answer

The equation that has real roots is option 3: \( 2x^2 - 7x + 5 = 0 \).

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