The greatest negative integer satisfying `x^(2)-4x-77 lt 0 and x^(2) gt 4` is equal to

Options

A. `-3`
B. `-5`
C. `-6`
D. none of these

Answer

To solve the problem, we need to find the greatest negative integer that satisfies the inequalities \(x^2 - 4x - 77 < 0\) and \(x^2 > 4\).

### Step 1: Solve the first inequality \(x^2 - 4x - 77 < 0\)

1. **Find the roots of the equation**:
We start by solving the equation \(x^2 - 4x - 77 = 0\) using the quadratic formula:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Here, \(a = 1\), \(b = -4\), and \(c = -77\).
\[ x = \frac{4 \pm \sqrt{(-4)^2 - 4 \cdot 1 \cdot (-77)}}{2 \cdot 1} \]
\[ = \frac{4 \pm \sqrt{16 + 308}}{2} \]
\[ = \frac{4 \pm \sqrt{324}}{2} \]
\[ = \frac{4 \pm 18}{2} \]
This gives us the roots:
\[ x = \frac{22}{2} = 11 \quad \text{and} \quad x = \frac{-14}{2} = -7 \]

2. **Determine the intervals**:
The roots divide the number line into intervals: \((-∞, -7)\), \((-7, 11)\), and \((11, ∞)\).
We test a point from each interval to see where the inequality holds.

- For \(x = -8\) (in \((-∞, -7)\)):
\[ (-8)^2 - 4(-8) - 77 = 64 + 32 - 77 = 19 \quad (\text{not satisfied}) \]
- For \(x = 0\) (in \((-7, 11)\)):
\[ 0^2 - 4(0) - 77 = -77 \quad (\text{satisfied}) \]
- For \(x = 12\) (in \((11, ∞)\)):
\[ 12^2 - 4(12) - 77 = 144 - 48 - 77 = 19 \quad (\text{not satisfied}) \]

Thus, the solution for the first inequality is:
\[ -7 < x < 11 \]

### Step 2: Solve the second inequality \(x^2 > 4\)

1. **Find the roots of the equation**:
We solve \(x^2 - 4 = 0\):
\[ x^2 = 4 \implies x = 2 \quad \text{and} \quad x = -2 \]

2. **Determine the intervals**:
The roots divide the number line into intervals: \((-∞, -2)\), \((-2, 2)\), and \((2, ∞)\).
We test a point from each interval to see where the inequality holds.

- For \(x = -3\) (in \((-∞, -2)\)):
\[ (-3)^2 > 4 \quad (\text{satisfied}) \]
- For \(x = 0\) (in \((-2, 2)\)):
\[ 0^2 > 4 \quad (\text{not satisfied}) \]
- For \(x = 3\) (in \((2, ∞)\)):
\[ 3^2 > 4 \quad (\text{satisfied}) \]

Thus, the solution for the second inequality is:
\[ x < -2 \quad \text{or} \quad x > 2 \]

### Step 3: Combine the solutions

Now we need to find the intersection of the two inequalities:
1. From the first inequality: \(-7 < x < 11\)
2. From the second inequality: \(x < -2\) or \(x > 2\)

The relevant part of the first inequality that intersects with the second is:
\[ -7 < x < -2 \]

### Step 4: Find the greatest negative integer

The integers in the interval \((-7, -2)\) are \(-6, -5, -4, -3\). The greatest negative integer in this set is:
\[ \boxed{-3} \]