Which of the following quadratic equations has real roots ?

Options

A. `4x ^(2)- 9x + 6=0`
B. `3x ^(2)-2x + 6=0`
C. `2x ^(2) - 7x + 6=0`
D. `x ^(2) - 2x + 2=0`

Answer

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

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

A quadratic equation has real roots if the discriminant \( D \) is greater than or equal to zero (\( D \geq 0 \)).

Let's analyze each option step by step:

### Step 1: Analyze Option A
Assume the quadratic equation is of the form \( 4x^2 + bx + 16 = 0 \).

1. Identify coefficients:
- \( a = 4 \)
- \( b = b \) (unknown)
- \( c = 16 \)

2. Calculate the discriminant:
\[ D = b^2 - 4ac = b^2 - 4 \cdot 4 \cdot 16 = b^2 - 256 \]

3. For real roots, we need:
\[ b^2 - 256 \geq 0 \implies b^2 \geq 256 \implies |b| \geq 16 \]

Since we do not have a specific value for \( b \), we cannot conclude definitively whether this option has real roots without knowing \( b \).

### Step 2: Analyze Option B
Assume the quadratic equation is of the form \( 3x^2 - 2x + 6 = 0 \).

1. Identify coefficients:
- \( a = 3 \)
- \( b = -2 \)
- \( c = 6 \)

2. Calculate the discriminant:
\[ D = (-2)^2 - 4 \cdot 3 \cdot 6 = 4 - 72 = -68 \]

Since \( D < 0 \), this option does not have real roots.

### Step 3: Analyze Option C
Assume the quadratic equation is of the form \( x^2 - 7x + 6 = 0 \).

1. Identify coefficients:
- \( a = 1 \)
- \( b = -7 \)
- \( c = 6 \)

2. Calculate the discriminant:
\[ D = (-7)^2 - 4 \cdot 1 \cdot 6 = 49 - 24 = 25 \]

Since \( D > 0 \), this option has real roots.

### Step 4: Analyze Option D
Assume the quadratic equation is of the form \( 2x^2 + 3x + 5 = 0 \).

1. Identify coefficients:
- \( a = 2 \)
- \( b = 3 \)
- \( c = 5 \)

2. Calculate the discriminant:
\[ D = 3^2 - 4 \cdot 2 \cdot 5 = 9 - 40 = -31 \]

Since \( D < 0 \), this option does not have real roots.

### Conclusion
The only quadratic equation among the options that has real roots is Option C.