Solve the following inequality.
x squared minus 15 x plus 54 greater than 0

To solve the inequality \( x^2 - 15x + 54 > 0 \), we first find the roots of the corresponding quadratic equation \( x^2 - 15x + 54 = 0 \) using the quadratic formula:

\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]

Here, \( a = 1 \), \( b = -15 \), and \( c = 54 \).

1. Calculate the discriminant:
   \[
   b^2 - 4ac = (-15)^2 - 4 \cdot 1 \cdot 54 = 225 - 216 = 9
   \]

2. Find the roots:
   \[
   x = \frac{15 \pm \sqrt{9}}{2 \cdot 1} = \frac{15 \pm 3}{2}
   \]
   This gives us:
   \[
   x_1 = \frac{18}{2} = 9 \quad \text{and} \quad x_2 = \frac{12}{2} = 6
   \]

Now we have the roots \( x = 6 \) and \( x = 9 \). The quadratic \( x^2 - 15x + 54 \) opens upwards (since the coefficient of \( x^2 \) is positive), so it is positive outside the interval defined by the roots.

The intervals to test are:
- \( (-\infty, 6) \)
- \( (6, 9) \)
- \( (9, \infty) \)

**Testing the intervals:**
1. For \( x < 6 \) (e.g., \( x = 0 \)):
   \[
   0^2 - 15 \cdot 0 + 54 = 54 > 0
   \]
2. For \( 6 < x < 9 \) (e.g., \( x = 7 \)):
   \[
   7^2 - 15 \cdot 7 + 54 = 49 - 105 + 54 = -2 < 0
   \]
3. For \( x > 9 \) (e.g., \( x = 10 \)):
   \[
   10^2 - 15 \cdot 10 + 54 = 100 - 150 + 54 = 4 > 0
   \]

Thus, the solution to the inequality \( x^2 - 15x + 54 > 0 \) is:

\[
(-\infty, 6) \cup (9, \infty)
\]

### Summary:
The solution to the inequality is \( x \in (-\infty, 6) \cup (9, \infty) \).

Would you like further details on this solution, or do you have any questions? Here are some related questions for you:

1. What are the implications of the roots in quadratic inequalities?
2. How can we graphically represent this inequality?
3. What would the solution be if the inequality was less than or equal to zero?
4. Can you provide examples of similar quadratic inequalities?
5. How does the discriminant affect the nature of the roots? 

**Tip:** Always check the intervals around the roots to determine where the quadratic is positive or negative.

Solve the following inequality: x squared minus 15 x plus 54 greater than 0

Explore the step-by-step solution to the inequality x² - 15x + 54 > 0. This guide covers key algebraic concepts such as finding roots and testing intervals, making it perfect for students in Grades 9-10.

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