The given expression is:

\[ 8 - x - 9x^2 \]

This is a quadratic expression in the form \( ax^2 + bx + c \), where \( a = -9 \), \( b = -1 \), and \( c = 8 \).

### Key Characteristics:

1. **Standard Form**: 
   \[ -9x^2 - x + 8 \]

2. **Degree**: The highest power of \( x \) is 2, so it is a quadratic polynomial.

3. **Coefficients**:
   - \( a = -9 \)
   - \( b = -1 \)
   - \( c = 8 \)

### Factoring (if possible):

To factor the quadratic expression, we look for two numbers that multiply to \( a \cdot c = -9 \cdot 8 = -72 \) and add up to \( b = -1 \). These numbers are not immediately apparent, suggesting that the quadratic expression may not factor neatly with rational numbers.

### Finding Roots (Solving \( 8 - x - 9x^2 = 0 \)):

To find the roots, we use the quadratic formula:

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

Substitute \( a = -9 \), \( b = -1 \), and \( c = 8 \):

\[ x = \frac{-(-1) \pm \sqrt{(-1)^2 - 4(-9)(8)}}{2(-9)} \]

Simplify the expression under the square root:

\[ x = \frac{1 \pm \sqrt{1 + 288}}{-18} \]
\[ x = \frac{1 \pm \sqrt{289}}{-18} \]
\[ x = \frac{1 \pm 17}{-18} \]

This gives two solutions:

\[ x = \frac{1 + 17}{-18} = \frac{18}{-18} = -1 \]
\[ x = \frac{1 - 17}{-18} = \frac{-16}{-18} = \frac{8}{9} \]

So, the roots of the quadratic equation \( 8 - x - 9x^2 = 0 \) are:

\[ x = -1 \quad \text{and} \quad x = \frac{8}{9} \]

### Vertex Form:

The vertex form of a quadratic function \( ax^2 + bx + c \) is:

\[ a(x - h)^2 + k \]

To find \( h \) and \( k \):

\[ h = -\frac{b}{2a} = -\frac{-1}{2(-9)} = \frac{1}{-18} = -\frac{1}{18} \]

Substitute \( x = -\frac{1}{18} \) into the original equation to find \( k \):

\[ k = 8 - \left(-\frac{1}{18}\right) - 9\left(-\frac{1}{18}\right)^2 \]
\[ k = 8 + \frac{1}{18} - 9 \cdot \frac{1}{324} \]
\[ k = 8 + \frac{1}{18} - \frac{1}{36} \]
\[ k = 8 + \frac{2}{36} - \frac{1}{36} \]
\[ k = 8 + \frac{1}{36} \]
\[ k = 8 + 0.0278 \]
\[ k \approx 8.0278 \]

Thus, the vertex form is:

\[ -9\left(x + \frac{1}{18}\right)^2 + 8.0278 \]

Would you like more details or have any questions?

### Suggested Questions:
1. How do you complete the square for a quadratic expression?
2. What is the significance of the discriminant in the quadratic formula?
3. Can you explain how to convert a quadratic expression into vertex form?
4. How do you determine the axis of symmetry for a quadratic function?
5. What are some real-world applications of quadratic equations?

### Tip:
Always check if a quadratic expression can be factored easily by looking for two numbers that multiply to \( ac \) and add up to \( b \). This can save time compared to using the quadratic formula.

Explore how to solve the quadratic expression 8 - x - 9x^2 step-by-step. This problem involves key concepts such as quadratic equations, vertex form, and the quadratic formula. Suitable for high school students studying algebra and quadratic equations.

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