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.