If `2a+3b+6c=0,` then prove that at least one root of the equation `a x^2+b x+c=0` lies in the interval (0,1).

Explanation

To prove that at least one root of the equation \( ax^2 + bx + c = 0 \) lies in the interval \( (0, 1) \) given that \( 2a + 3b + 6c = 0 \), we can follow these steps:

### Step 1: Define the function
Let \( f(x) = ax^2 + bx + c \). We want to analyze this function over the interval \( (0, 1) \).

### Step 2: Evaluate \( f(0) \) and \( f(1) \)
Calculate the values of the function at the endpoints of the interval:
- \( f(0) = a(0)^2 + b(0) + c = c \)
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