Each of a and b can take values of 1 or 2 with equal probability . The probability that the equation `ax^(2)+bx+1=0` has real roots, is equal to

Options

A. `1/2`
B. `1/4`
C. `1/8`
D. `1/16`

Answer

To solve the problem, we need to determine the probability that the quadratic equation \( ax^2 + bx + 1 = 0 \) has real roots, given that \( a \) and \( b \) can take values of 1 or 2 with equal probability.

### Step 1: Identify the condition for real roots
For a quadratic equation \( ax^2 + bx + c = 0 \) to have real roots, the discriminant \( D \) must be greater than or equal to zero. The discriminant is given by:
\[ D = b^2 - 4ac \]
In our case, \( c = 1 \), so the discriminant becomes:
\[ D = b^2 - 4a \]
We need to find when \( D \geq 0 \):
\[ b^2 - 4a \geq 0 \]

### Step 2: List the possible values for \( a \) and \( b \)
Since \( a \) and \( b \) can each take values of 1 or 2, we can list the possible combinations of \( (a, b) \):
1. \( (1, 1) \)
2. \( (1, 2) \)
3. \( (2, 1) \)
4. \( (2, 2) \)

### Step 3: Calculate the discriminant for each combination
Now, we will calculate \( D \) for each combination to check if it is non-negative:
1. For \( (1, 1) \):
\[ D = 1^2 - 4 \cdot 1 = 1 - 4 = -3 \quad (\text{not real roots}) \]
2. For \( (1, 2) \):
\[ D = 2^2 - 4 \cdot 1 = 4 - 4 = 0 \quad (\text{real roots}) \]
3. For \( (2, 1) \):
\[ D = 1^2 - 4 \cdot 2 = 1 - 8 = -7 \quad (\text{not real roots}) \]
4. For \( (2, 2) \):
\[ D = 2^2 - 4 \cdot 2 = 4 - 8 = -4 \quad (\text{not real roots}) \]

### Step 4: Count the successful outcomes
From the calculations, we find that the only combination that gives real roots is \( (1, 2) \). Thus, there is 1 successful outcome.

### Step 5: Calculate the total number of outcomes
Since \( a \) and \( b \) can each take 2 values (1 or 2), the total number of combinations is:
\[ 2 \times 2 = 4 \]

### Step 6: Calculate the probability
The probability \( P \) that the equation has real roots is given by the ratio of successful outcomes to total outcomes:
\[ P = \frac{\text{Number of successful outcomes}}{\text{Total outcomes}} = \frac{1}{4} \]

### Final Answer
Thus, the probability that the equation \( ax^2 + bx + 1 = 0 \) has real roots is:
\[ \frac{1}{4} \]