If the roots of the equation `x ^(2)-ax -b=0(a,b, in R)` are both lying between `-2 and 2,` then :

Options

A. `|a| lt 2 - (b)/(2)`
B. ` |a| gt 2 -(b)/(2)`
C. `|a| lt 4`
D. ` |a| gt (b)/(2) -2`

Answer

To solve the problem, we need to analyze the quadratic equation given by \(x^2 - ax - b = 0\) and determine the conditions under which both roots lie between -2 and 2.

### Step 1: Understand the conditions for the roots
For the roots of the quadratic equation to be real, the discriminant must be non-negative. The discriminant \(D\) for the equation \(x^2 - ax - b = 0\) is given by:

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

Since we want the roots to be real, we need:

\[ a^2 - 4b \geq 0 \quad \text{(1)} \]

### Step 2: Roots lying between -2 and 2
Let the roots of the quadratic equation be \(r_1\) and \(r_2\). For both roots to lie between -2 and 2, we can use Vieta's formulas, which state that:

- The sum of the roots \(r_1 + r_2 = a\)
- The product of the roots \(r_1 r_2 = -b\)

From the condition that both roots are between -2 and 2, we can derive the following inequalities:

1. The sum of the roots must be less than 4:
\[ r_1 + r_2 < 4 \implies a < 4 \quad \text{(2)} \]

2. The sum of the roots must be greater than -4:
\[ r_1 + r_2 > -4 \implies a > -4 \quad \text{(3)} \]

3. The product of the roots must be greater than 4:
\[ r_1 r_2 > -4 \implies -b > -4 \implies b < 4 \quad \text{(4)} \]

4. The product of the roots must be greater than 0:
\[ r_1 r_2 < 4 \implies -b < 4 \implies b > -4 \quad \text{(5)} \]

### Step 3: Combine the inequalities
From inequalities (2) and (3), we have:

\[ -4 < a < 4 \]

From inequalities (4) and (5), we have:

\[ -4 < b < 4 \]

### Step 4: Analyze the conditions for \(a\) and \(b\)
Now we need to combine these conditions with the discriminant condition (1):

\[ a^2 \geq 4b \]

### Conclusion
The conditions we have derived are:

1. \(a^2 \geq 4b\)
2. \(-4 < a < 4\)
3. \(-4 < b < 4\)

This means that the values of \(a\) and \(b\) must satisfy these inequalities for both roots of the quadratic equation to lie between -2 and 2.