If a, b, c `in R` and 2a + 3b + 6c = 0, then the equation `ax^(2) + bx + c = 0` has

Options

A. at least one root in [0,1]
B. at least one root in [2,3]
C. at least one root in [-1,0]
D. at least one root in `[-infty, 1]`

Answer

To determine the nature of the roots of the quadratic equation \( ax^2 + bx + c = 0 \) given the condition \( 2a + 3b + 6c = 0 \), we can follow these steps:

### Step 1: Understand the Condition
We start with the equation \( 2a + 3b + 6c = 0 \). This condition relates the coefficients \( a \), \( b \), and \( c \) of the quadratic equation.

### Step 2: Rearranging the Condition
We can rearrange the condition to express \( c \) in terms of \( a \) and \( b \):
\[ 6c = -2a - 3b \implies c = -\frac{2a + 3b}{6} \]

### Step 3: Substitute \( c \) into the Quadratic Equation
Now, we substitute this expression for \( c \) back into the quadratic equation:
\[ ax^2 + bx + \left(-\frac{2a + 3b}{6}\right) = 0 \]
This simplifies to:
\[ ax^2 + bx - \frac{2a + 3b}{6} = 0 \]

### Step 4: Multiply through by 6 to Eliminate the Fraction
To eliminate the fraction, we can multiply the entire equation by 6:
\[ 6ax^2 + 6bx - (2a + 3b) = 0 \]
This gives us:
\[ 6ax^2 + 6bx - 2a - 3b = 0 \]

### Step 5: Apply the Discriminant Condition
The nature of the roots of a quadratic equation \( Ax^2 + Bx + C = 0 \) can be determined using the discriminant \( D = B^2 - 4AC \). Here, \( A = 6a \), \( B = 6b \), and \( C = -(2a + 3b) \).

Calculating the discriminant:
\[ D = (6b)^2 - 4(6a)(-(2a + 3b)) \]
\[ D = 36b^2 + 24a(2a + 3b) \]
\[ D = 36b^2 + 48a^2 + 72ab \]

### Step 6: Analyze the Discriminant
Since \( 36b^2 + 48a^2 + 72ab \) is a sum of squares and products, it is always non-negative. Therefore, \( D \geq 0 \).

### Conclusion
Since the discriminant \( D \) is non-negative, the quadratic equation \( ax^2 + bx + c = 0 \) has at least one real root.