First, given the quadratic equation [tex]\( y = -2x^2 - 9x + 9 \)[/tex], we need to find the [tex]\( x \)[/tex]-intercepts, where the graph of the equation crosses the x-axis. The [tex]\( x \)[/tex]-intercepts occur where [tex]\( y = 0 \)[/tex].
So, we set the equation to zero:
[tex]\[ -2x^2 - 9x + 9 = 0 \][/tex]
We can solve this quadratic equation either using the quadratic formula [tex]\( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)[/tex] or graphing calculator methods.
### Method 1: Using the CALC-Zero Feature of a Graphing Calculator
1. Enter the Equation:
- Enter [tex]\( y = -2x^2 - 9x + 9 \)[/tex] into the graphing calculator.
2. Graph the Equation:
- Graph the equation to visualize where it crosses the x-axis.
3. Find the Zeros:
- Use the `CALC` menu to select the `Zero` feature.
- For the first intercept, move the cursor near where the graph crosses the x-axis (right intercept) and follow the prompts to find the zero.
- For the second intercept, move the cursor near the other crossing (left intercept) and find the zero.
### Method 2: Using the CALC-Intersect Feature of a Graphing Calculator
1. Enter the Equations:
- Enter [tex]\( Y_1 = -2x^2 - 9x + 9 \)[/tex] into your graphing calculator.
- Enter [tex]\( Y_2 = 0 \)[/tex] (which is simply the x-axis).
2. Graph the Equations:
- Graph both [tex]\( Y_1 \)[/tex] and [tex]\( Y_2 \)[/tex] on the same set of axes.
3. Find the Intersection Points:
- Use the `CALC` menu to select the `Intersect` feature.
- For the first intercept (right intercept), move the cursor near the intersection point on the right and follow the prompts to determine the x-coordinate of the intersection.
- For the second intercept (left intercept), move the cursor near the intersection point on the left and find the x-coordinate.
After using one of the methods, we find the x-intercepts are approximately:
- [tex]\( x = 0.842 \)[/tex]
- [tex]\( x = -5.342 \)[/tex]
### Solution:
The [tex]\( x \)[/tex]-intercepts, ordered with the larger value first, are:
[tex]\[ (0.842, 0) \][/tex]
[tex]\[ (-5.342, 0) \][/tex]
