The discriminant of the quadratic `(2x+1)^(2)+(3x+2)^(2)+(4x+3)^(2)+….n` terms `=0, AA n gt 3, x in R` is

Options

A. positive
B. zero
C. negative
D. depends on n

Answer

To solve the problem, we need to analyze the quadratic expression given by the sum of squares of linear functions. The expression is:

\[ (2x + 1)^2 + (3x + 2)^2 + (4x + 3)^2 + \ldots + (n + 1)^2 \]

We need to find the discriminant of this quadratic expression and determine its nature based on the number of terms \( n \) (where \( n > 3 \)).

### Step 1: Expand Each Term
First, we expand each term in the quadratic expression:

\[ (2x + 1)^2 = 4x^2 + 4x + 1 \]
\[ (3x + 2)^2 = 9x^2 + 12x + 4 \]
\[ (4x + 3)^2 = 16x^2 + 24x + 9 \]
\[ \vdots \]
\[ ((n + 1)x + n)^2 = (n + 1)^2x^2 + 2(n + 1)nx + n^2 \]

### Step 2: Combine Like Terms
Next, we combine the coefficients of \( x^2 \), \( x \), and the constant terms from all \( n \) terms.

The coefficient of \( x^2 \) (denoted as \( a \)) is the sum of the squares of the coefficients of \( x \):
\[ a = 4 + 9 + 16 + \ldots + (n + 1)^2 \]

This can be expressed as:
\[ a = \sum_{k=2}^{n} k^2 = \frac{n(n + 1)(2n + 1)}{6} - 1 \]

The coefficient of \( x \) (denoted as \( b \)) is:
\[ b = 4 + 12 + 24 + \ldots + 2(n + 1)n \]

This can be expressed as:
\[ b = 2 \sum_{k=2}^{n} k(k - 1) = n(n + 1) \]

The constant term (denoted as \( c \)) is:
\[ c = 1 + 4 + 9 + \ldots + n^2 = \frac{n(n + 1)(2n + 1)}{6} \]

### Step 3: Calculate the Discriminant
The discriminant \( D \) of a quadratic equation \( ax^2 + bx + c = 0 \) is given by:
\[ D = b^2 - 4ac \]

Substituting our expressions for \( a \), \( b \), and \( c \):
\[ D = (n(n + 1))^2 - 4\left(\frac{n(n + 1)(2n + 1)}{6} - 1\right)\left(\frac{n(n + 1)(2n + 1)}{6}\right) \]

### Step 4: Analyze the Discriminant
To determine the nature of the discriminant, we need to analyze \( D \) based on \( n \):
- If \( D > 0 \), the quadratic has two distinct real roots.
- If \( D = 0 \), the quadratic has exactly one real root.
- If \( D < 0 \), the quadratic has no real roots.

From the analysis, we find that the discriminant depends on the value of \( n \). For \( n > 3 \), it can be shown that \( D \) will not be consistently positive or negative, indicating that the nature of the roots depends on \( n \).

### Final Answer
Thus, the discriminant of the quadratic expression is dependent on \( n \).