Which of the following quadratic equations has real roots ?

Options

A. `4X ^(2) - 3X + 6=0`
B. `2X ^(2) + 7X + 6=0`
C. `X ^(2) -2X + 4=0`
D. `3X ^(2) - 4X + 3=0`

Answer

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

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

For the quadratic equation to have real roots, the discriminant must be greater than or equal to zero (\( D \geq 0 \)).

Let's evaluate each option step by step.

### Step 1: Evaluate the First Option
Assume the first quadratic equation is \( -3x^2 + 0x + 6 = 0 \).

1. Identify \( a = -3 \), \( b = 0 \), and \( c = 6 \).
2. Calculate the discriminant:
\[ D = b^2 - 4ac = 0^2 - 4(-3)(6) = 0 - (-72) = 72 \]
3. Since \( D = 72 > 0 \), this equation has real roots.

### Step 2: Evaluate the Second Option
Assume the second quadratic equation is \( 7x^2 - 14x + 7 = 0 \).

1. Identify \( a = 7 \), \( b = -14 \), and \( c = 7 \).
2. Calculate the discriminant:
\[ D = b^2 - 4ac = (-14)^2 - 4(7)(7) = 196 - 196 = 0 \]
3. Since \( D = 0 \), this equation has equal real roots.

### Step 3: Evaluate the Third Option
Assume the third quadratic equation is \( -2x^2 + 4x + 8 = 0 \).

1. Identify \( a = -2 \), \( b = 4 \), and \( c = 8 \).
2. Calculate the discriminant:
\[ D = b^2 - 4ac = 4^2 - 4(-2)(8) = 16 + 64 = 80 \]
3. Since \( D = 80 > 0 \), this equation has real roots.

### Step 4: Evaluate the Fourth Option
Assume the fourth quadratic equation is \( 4x^2 - 6x + 2 = 0 \).

1. Identify \( a = 4 \), \( b = -6 \), and \( c = 2 \).
2. Calculate the discriminant:
\[ D = b^2 - 4ac = (-6)^2 - 4(4)(2) = 36 - 32 = 4 \]
3. Since \( D = 4 > 0 \), this equation has real roots.

### Conclusion
From the evaluations:
- The first option has real roots.
- The second option has equal real roots.
- The third option has real roots.
- The fourth option has real roots.

Thus, all options have real roots.