Let `f(x)=ax^(2)+bx+c`, `a ne 0`, `a`, `b`, `c in I`. Suppose that `f(1)=0`, `50 lt f(7) lt 60 ` and `70 lt f(8) lt 80`.
Number of integral values of `x` for which `f(x) lt 0` is

Options

A. `0`
B. `1`
C. `2`
D. `3`

Explanation

To solve the problem, we need to analyze the quadratic function \( f(x) = ax^2 + bx + c \) given the conditions provided.

### Step 1: Use the condition \( f(1) = 0 \)
Since \( f(1) = 0 \), we can substitute \( x = 1 \) into the function:
\[ f(1) = a(1)^2 + b(1) + c = a + b + c = 0 \]
From this, we can express \( c \) in terms of \( a \) and \( b \):
\[ c = -a - b \quad \text{(Equation 1)} \]

### Step 2: Use the condition \( 50 < f(7) < 60 \)
Next, we substitute \( x = 7 \) into the function:
\[ f(7) = a(7)^2 + b(7) + c = 49a + 7b + c \]
Substituting \( c \) from Equation 1:
\[ f(7) = 49a + 7b - a - b = 48a + 6b \]
We have the inequality:
\[ 50 < 48a + 6b < 60 \]
This can be split into two inequalities:
1. \( 48a + 6b > 50 \)
2. \( 48a + 6b < 60 \)

### Step 3: Simplify the inequalities
Dividing the entire inequalities by 6:
1. \( 8a + b > \frac{25}{3} \approx 8.33 \)
2. \( 8a + b < 10 \)

### Step 4: Analyze the inequalities
From the inequalities \( 8a + b > 8.33 \) and \( 8a + b < 10 \), we can conclude:
\[ 8.33 < 8a + b < 10 \quad \text{(Equation 2)} \]

### Step 5: Use the condition \( 70 < f(8) < 80 \)
Now, we substitute \( x = 8 \):
\[ f(8) = a(8)^2 + b(8) + c = 64a + 8b + c \]
Substituting \( c \) from Equation 1:
\[ f(8) = 64a + 8b - a - b = 63a + 7b \]
We have the inequality:
\[ 70 < 63a + 7b < 80 \]
Dividing the entire inequalities by 7:
1. \( 9 < 9a + b < \frac{80}{7} \approx 11.43 \)

### Step 6: Analyze the inequalities
From the inequalities \( 9 < 9a + b < 11.43 \), we can conclude:
\[ 9 < 9a + b < 11.43 \quad \text{(Equation 3)} \]

### Step 7: Solve the system of inequalities
Now we have two systems of inequalities:
1. From Equation 2: \( 8.33 < 8a + b < 10 \)
2. From Equation 3: \( 9 < 9a + b < 11.43 \)

We can express \( b \) in terms of \( a \):
From \( 8a + b < 10 \):
\[ b < 10 - 8a \]
From \( 9a + b > 9 \):
\[ b > 9 - 9a \]

### Step 8: Find integer values of \( a \) and \( b \)
We can solve for \( a \) and \( b \) using the inequalities:
1. \( 8.33 < 8a + b < 10 \)
2. \( 9 < 9a + b < 11.43 \)

By solving these inequalities, we can find possible integer values for \( a \) and \( b \).

### Step 9: Determine the quadratic function
Once we find \( a \) and \( b \), we can substitute back to find \( c \) using Equation 1.

### Step 10: Analyze the roots of the quadratic function
Using the quadratic \( f(x) = ax^2 + bx + c \), we can determine the roots and the intervals where \( f(x) < 0 \).

### Step 11: Count the integral values of \( x \)
Finally, we will count the integral values of \( x \) for which \( f(x) < 0 \).