The task is to solve the quadratic inequality:
\[
x^2 - 9x + 20 > 0
\]
Step-by-Step Solution:
####
1. Factor the quadratic expression:
The given quadratic expression is \( x^2 - 9x + 20 \). To factor it, we need to find two numbers that:
- Multiply to give the constant term \( 20 \).
- Add up to give the coefficient of \( x \), which is \( -9 \).
The two numbers that satisfy these conditions are \( -4 \) and \( -5 \) because:
\[
-4 \times -5 = 20 \quad \text{and} \quad -4 + (-5) = -9
\]
Thus, we can factor the quadratic expression as:
\[
x^2 - 9x + 20 = (x - 4)(x - 5)
\]
So the inequality becomes:
\[
(x - 4)(x - 5) > 0
\]
####
2. Find the critical points:
The critical points are the values of \( x \) that make the expression equal to zero. These are the roots of the equation:
\[
(x - 4)(x - 5) = 0
\]
Setting each factor to zero gives:
\[
x - 4 = 0 \quad \Rightarrow \quad x = 4
\]
\[
x - 5 = 0 \quad \Rightarrow \quad x = 5
\]
So, the critical points are \( x = 4 \) and \( x = 5 \).
####
3. Determine the intervals:
The critical points divide the real number line into three intervals:
\[
(-\infty, 4), \quad (4, 5), \quad (5, \infty)
\]
We need to test the sign of the expression \( (x - 4)(x - 5) \) in each interval.
####
4. Test the sign in each interval:
-
Interval \( (-\infty, 4) \):
Choose a test point, say \( x = 0 \):
\[
(0 - 4)(0 - 5) = (-4)(-5) = 20 \quad (\text{positive})
\]
-
Interval \( (4, 5) \):
Choose a test point, say \( x = 4.5 \):
\[
(4.5 - 4)(4.5 - 5) = (0.5)(-0.5) = -0.25 \quad (\text{negative})
\]
-
Interval \( (5, \infty) \):
Choose a test point, say \( x = 6 \):
\[
(6 - 4)(6 - 5) = (2)(1) = 2 \quad (\text{positive})
\]
####
5. Determine where the inequality holds:
The inequality \( (x - 4)(x - 5) > 0 \) is satisfied when the expression is positive. From the sign analysis:
- The expression is positive in the intervals \( (-\infty, 4) \) and \( (5, \infty) \).
####
6. Write the solution:
The solution to the inequality \( x^2 - 9x + 20 > 0 \) is:
\[
x < 4 \quad \text{or} \quad x > 5
\]
In interval notation, this is:
\[
(-\infty, 4) \cup (5, \infty)
\]
Final Answer:
\[
\boxed{x < 4 \text{ or } x > 5}
\]
Parent Tip: Review the logic above to help your child master the concept of solving quadratic inequalities worksheet.
