Sure! Let's solve the quadratic inequality [tex]\( x^2 - 4x - 21 \leq 0 \)[/tex].
### Step 1: Identify the roots of the quadratic equation
First, we need to solve for the roots of the corresponding quadratic equation [tex]\( x^2 - 4x - 21 = 0 \)[/tex]. These roots are the values of [tex]\( x \)[/tex] where the quadratic function intersects the x-axis.
The roots of the quadratic equation [tex]\( x^2 - 4x - 21 = 0 \)[/tex] are:
[tex]\[ x = -3 \text{ and } x = 7 \][/tex]
### Step 2: Determine the intervals
The roots [tex]\(-3\)[/tex] and [tex]\(7\)[/tex] divide the number line into three intervals:
1. [tex]\( (-\infty, -3) \)[/tex]
2. [tex]\( (-3, 7) \)[/tex]
3. [tex]\( (7, \infty) \)[/tex]
### Step 3: Test the sign of the quadratic expression in each interval
To determine in which intervals the quadratic expression [tex]\( x^2 - 4x - 21 \)[/tex] is less than or equal to zero, we test points within each interval.
1. For the interval [tex]\( (-\infty, -3) \)[/tex], choose [tex]\( x = -4 \)[/tex]:
[tex]\[ (-4)^2 - 4(-4) - 21 = 16 + 16 - 21 = 11 \][/tex]
The expression is positive, so [tex]\( x^2 - 4x - 21 > 0 \)[/tex] in this interval.
2. For the interval [tex]\( (-3, 7) \)[/tex], choose [tex]\( x = 0 \)[/tex]:
[tex]\[ (0)^2 - 4(0) - 21 = -21 \][/tex]
The expression is negative, so [tex]\( x^2 - 4x - 21 < 0 \)[/tex] in this interval.
3. For the interval [tex]\( (7, \infty) \)[/tex], choose [tex]\( x = 8 \)[/tex]:
[tex]\[ (8)^2 - 4(8) - 21 = 64 - 32 - 21 = 11 \][/tex]
The expression is positive, so [tex]\( x^2 - 4x - 21 > 0 \)[/tex] in this interval.
### Step 4: Include or exclude the roots
Since we need to solve the inequality [tex]\( x^2 - 4x - 21 \leq 0 \)[/tex], we include the roots since the expression is equal to zero at these points.
### Solution
Therefore, the solution to the quadratic inequality [tex]\( x^2 - 4x - 21 \leq 0 \)[/tex] is the interval where the expression is less than or equal to zero, including the roots:
[tex]\[ -3 \leq x \leq 7 \][/tex]
So, the solution is:
[tex]\[ x \in [-3, 7] \][/tex]
