The number of integral values of m for which the equation `(1+m^(2)) x^(2) - 2(1+3m)x+(1+8m) = 0`, has no real roots is

Options

A. 1
B. 2
C. 3
D. infinitely many

Explanation

To determine the number of integral values of \( m \) for which the quadratic equation

\[ (1+m^2)x^2 - 2(1+3m)x + (1+8m) = 0 \]

has no real roots, we need to analyze the discriminant of the quadratic equation. A quadratic equation has no real roots if its discriminant is less than zero.

### Step 1: Identify coefficients
The coefficients of the quadratic equation are:
- \( a = 1 + m^2 \)
- \( b = -2(1 + 3m) \)
- \( c = 1 + 8m \)

### Step 2: Write the discriminant
The discriminant \( D \) of a quadratic equation \( ax^2 + bx + c = 0 \) is given by:

\[ D = b^2 - 4ac \]

Substituting our coefficients into the discriminant formula, we have:

\[ D = [-2(1 + 3m)]^2 - 4(1 + m^2)(1 + 8m) \]

### Step 3: Simplify the discriminant
Calculating \( D \):

1. Calculate \( b^2 \):
\[ b^2 = 4(1 + 3m)^2 = 4(1 + 6m + 9m^2) = 4 + 24m + 36m^2 \]

2. Calculate \( 4ac \):
\[ 4ac = 4(1 + m^2)(1 + 8m) = 4[(1 + m^2) + 8m(1 + m^2)] = 4(1 + m^2 + 8m + 8m^3) \]
\[ = 4 + 4m^2 + 32m + 32m^3 \]

3. Now substitute back into the discriminant:
\[ D = (4 + 24m + 36m^2) - (4 + 4m^2 + 32m + 32m^3) \]
\[ = 4 + 24m + 36m^2 - 4 - 4m^2 - 32m - 32m^3 \]
\[ = 36m^2 - 4m^2 + 24m - 32m - 32m^3 \]
\[ = -32m^3 + 32m^2 - 8m \]

### Step 4: Set the discriminant less than zero
To find when there are no real roots, we set the discriminant less than zero:

\[ -32m^3 + 32m^2 - 8m < 0 \]

### Step 5: Factor the inequality
Factoring out \(-8m\):

\[ -8m(4m^2 - 4m + 1) < 0 \]

### Step 6: Analyze the quadratic factor
The quadratic \( 4m^2 - 4m + 1 \) can be analyzed using the discriminant:

\[ D = (-4)^2 - 4 \cdot 4 \cdot 1 = 16 - 16 = 0 \]

Since the discriminant is zero, \( 4m^2 - 4m + 1 \) has a double root at:

\[ m = \frac{4}{2 \cdot 4} = \frac{1}{2} \]

### Step 7: Determine the sign of the quadratic
The quadratic \( 4m^2 - 4m + 1 \) is always positive (it opens upwards and has a double root). Therefore, the sign of the expression \(-8m(4m^2 - 4m + 1)\) depends solely on \( -8m \).

### Step 8: Solve for \( m \)
Thus, we need:

\[ -8m < 0 \implies m > 0 \]

### Step 9: Count integral values
The integral values of \( m \) that satisfy \( m > 0 \) are \( 1, 2, 3, \ldots \), which are infinitely many.

### Final Answer
The number of integral values of \( m \) for which the equation has no real roots is **infinitely many**.

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