Solve `x^(2) - 4x - 21 ge 0` by algebric method and graphical method.

Answer

To solve the quadratic inequality \( x^2 - 4x - 21 \geq 0 \) using both algebraic and graphical methods, we can follow these steps:

### Algebraic Method

1. **Rewrite the Inequality**:
We start with the inequality:
\[ x^2 - 4x - 21 \geq 0 \]

2. **Factor the Quadratic Expression**:
We need to factor the quadratic expression \( x^2 - 4x - 21 \). We look for two numbers that multiply to \(-21\) (the constant term) and add to \(-4\) (the coefficient of \(x\)). The numbers \(-7\) and \(3\) satisfy these conditions:
\[ x^2 - 4x - 21 = (x - 7)(x + 3) \]

3. **Set the Factors to Zero**:
To find the critical points, we set each factor equal to zero:
\[ x - 7 = 0 \quad \Rightarrow \quad x = 7 \]
\[ x + 3 = 0 \quad \Rightarrow \quad x = -3 \]

4. **Determine Intervals**:
The critical points divide the number line into intervals. We will test the sign of the expression \( (x - 7)(x + 3) \) in each interval:
- Interval 1: \( (-\infty, -3) \)
- Interval 2: \( (-3, 7) \)
- Interval 3: \( (7, \infty) \)

5. **Test Each Interval**:
- For \( x < -3 \) (e.g., \( x = -4 \)):
\[ (-4 - 7)(-4 + 3) = (-11)(-1) = 11 \quad (\text{positive}) \]
- For \( -3 < x < 7 \) (e.g., \( x = 0 \)):
\[ (0 - 7)(0 + 3) = (-7)(3) = -21 \quad (\text{negative}) \]
- For \( x > 7 \) (e.g., \( x = 8 \)):
\[ (8 - 7)(8 + 3) = (1)(11) = 11 \quad (\text{positive}) \]

6. **Write the Solution**:
The expression is non-negative in the intervals where we found positive values and at the critical points:
\[ x \in (-\infty, -3] \cup [7, \infty) \]

### Graphical Method

1. **Graph the Quadratic Function**:
We can graph the function \( f(x) = x^2 - 4x - 21 \). The roots of the function are at \( x = -3 \) and \( x = 7 \).

2. **Plot Points**:
We can calculate some values of \( f(x) \) to help plot the graph:
- \( f(-4) = 11 \)
- \( f(-3) = 0 \)
- \( f(0) = -21 \)
- \( f(1) = -24 \)
- \( f(7) = 0 \)
- \( f(8) = 11 \)

3. **Draw the Parabola**:
The parabola opens upwards (since the coefficient of \( x^2 \) is positive). The points plotted will help us visualize the graph.

4. **Identify Regions**:
The graph will show that the function is above the x-axis (non-negative) in the intervals \( (-\infty, -3] \) and \( [7, \infty) \).

5. **Conclusion**:
The solution to the inequality \( x^2 - 4x - 21 \geq 0 \) is:
\[ x \in (-\infty, -3] \cup [7, \infty) \]