Let a, b, c be non zero numbers such that
`int_(0)^(3)sqrt(x^(2)+x+1)(ax^(2)+bx+c)dx=int_(0)^(5)sqrt(1+x^(2)+x)(ax^(2)+bx+c)dx`. Then the quadratic equation `ax^(2)+bx+c=0` has

Options

A. no root in (0, 3)
B. at least one root in (3, 5)
C. a double root in (0, 3)
D. two imaginary roots

Answer

To solve the given problem, we need to analyze the equality of the two integrals and apply Rolle's theorem. Here’s a step-by-step solution:

### Step 1: Understand the given integrals
We have two integrals:
1. \( I_1 = \int_{0}^{3} \sqrt{x^2 + x + 1} (ax^2 + bx + c) \, dx \)
2. \( I_2 = \int_{0}^{5} \sqrt{1 + x^2 + x} (ax^2 + bx + c) \, dx \)

### Step 2: Set the integrals equal to each other
From the problem statement, we know that:
\[ I_1 = I_2 \]

### Step 3: Define the function
Let:
\[ f(x) = \int_{0}^{x} \sqrt{x^2 + x + 1} (ax^2 + bx + c) \, dx \]
Then, we can rewrite our integrals as:
\[ f(3) = \int_{0}^{3} \sqrt{x^2 + x + 1} (ax^2 + bx + c) \, dx \]
and
\[ f(5) = \int_{0}^{5} \sqrt{x^2 + x + 1} (ax^2 + bx + c) \, dx \]

### Step 4: Apply Rolle's Theorem
Since \( f(3) = f(5) \), by Rolle's theorem, there exists at least one \( c \) in the interval \( (3, 5) \) such that:
\[ f'(c) = 0 \]

### Step 5: Differentiate the function
To find \( f'(x) \), we apply the Fundamental Theorem of Calculus:
\[ f'(x) = \sqrt{x^2 + x + 1} (ax^2 + bx + c) \]

### Step 6: Set the derivative to zero
According to Rolle's theorem:
\[ f'(c) = \sqrt{c^2 + c + 1} (ac^2 + bc + c) = 0 \]
Since \( \sqrt{c^2 + c + 1} \) is always positive (as it is the square root of a positive quantity), we must have:
\[ ac^2 + bc + c = 0 \]

### Step 7: Factor the equation
We can factor out \( c \):
\[ c(ac + b + 1) = 0 \]
Since \( c \neq 0 \) (as stated in the problem), we have:
\[ ac + b + 1 = 0 \]

### Step 8: Conclusion about the roots of the quadratic equation
The quadratic equation \( ax^2 + bx + c = 0 \) must have at least one root in the interval \( (3, 5) \). This means that at least one root of the quadratic equation lies between 3 and 5.

### Final Answer
Thus, the quadratic equation \( ax^2 + bx + c = 0 \) has at least one root in the interval \( (3, 5) \).

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