The sum of two non integral roots of `|{:(x,2,5),(3,x,3),(5,4,x):}|=0` is

Options

A. 5
B. `-5`
C. `-18`
D. none of these

Answer

To solve the given determinant equation and find the sum of the two non-integral roots, we will follow these steps:

### Step 1: Set Up the Determinant
We start with the determinant given in the problem:
\[ \begin{vmatrix} x & 2 & 5 \\ 3 & x & 3 \\ 5 & 4 & x \end{vmatrix} = 0 \]

### Step 2: Expand the Determinant
We will expand the determinant using the method of cofactor expansion. The determinant can be expanded as follows:
\[ D = x \begin{vmatrix} x & 3 \\ 4 & x \end{vmatrix} - 2 \begin{vmatrix} 3 & 3 \\ 5 & x \end{vmatrix} + 5 \begin{vmatrix} 3 & x \\ 5 & 4 \end{vmatrix} \]

Calculating each of these 2x2 determinants:
1. \( \begin{vmatrix} x & 3 \\ 4 & x \end{vmatrix} = x^2 - 12 \)
2. \( \begin{vmatrix} 3 & 3 \\ 5 & x \end{vmatrix} = 3x - 15 \)
3. \( \begin{vmatrix} 3 & x \\ 5 & 4 \end{vmatrix} = 12 - 5x \)

Substituting these back into the determinant:
\[ D = x(x^2 - 12) - 2(3x - 15) + 5(12 - 5x) \]

### Step 3: Simplify the Expression
Now we simplify the expression:
\[ D = x^3 - 12x - 6x + 30 + 60 - 25x \]
Combining like terms:
\[ D = x^3 - 43x + 90 \]

### Step 4: Set the Polynomial Equal to Zero
We set the polynomial equal to zero:
\[ x^3 - 43x + 90 = 0 \]

### Step 5: Find One Root Using the Rational Root Theorem
We can use the Rational Root Theorem to find at least one root. Testing \( x = 5 \):
\[ 5^3 - 43(5) + 90 = 125 - 215 + 90 = 0 \]
Thus, \( x = 5 \) is a root.

### Step 6: Factor the Polynomial
We factor the polynomial using synthetic division or polynomial long division:
\[ x^3 - 43x + 90 = (x - 5)(x^2 + 5x + 18) \]

### Step 7: Solve the Quadratic Equation
Now we solve the quadratic equation \( x^2 + 5x + 18 = 0 \) using the quadratic formula:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} = \frac{-5 \pm \sqrt{5^2 - 4 \cdot 1 \cdot 18}}{2 \cdot 1} \]
Calculating the discriminant:
\[ b^2 - 4ac = 25 - 72 = -47 \]
Since the discriminant is negative, the roots are non-real (complex).

### Step 8: Sum of the Non-Integral Roots
The sum of the roots of the quadratic equation \( x^2 + 5x + 18 = 0 \) is given by:
\[ \text{Sum} = -\frac{b}{a} = -\frac{5}{1} = -5 \]

### Final Answer
The sum of the two non-integral roots is:
\[ \boxed{-5} \]