Let's break down the solution to each part of the question step-by-step.
### 1. Finding the Solutions of the Equation [tex]\( x^2 - 2x - 4 = -3x + 9 \)[/tex]
To find the solutions, we first set [tex]\( x^2 - 2x - 4 \)[/tex] equal to [tex]\(-3x + 9\)[/tex].
[tex]\[ x^2 - 2x - 4 = -3x + 9 \][/tex]
Rearrange the equation to set it to zero:
[tex]\[ x^2 - 2x - 4 + 3x - 9 = 0 \][/tex]
Combine like terms:
[tex]\[ x^2 + x - 13 = 0 \][/tex]
Now, solve for [tex]\( x \)[/tex] using the quadratic formula [tex]\( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)[/tex], where [tex]\( a = 1 \)[/tex], [tex]\( b = 1 \)[/tex], and [tex]\( c = -13 \)[/tex]:
[tex]\[ x = \frac{-1 \pm \sqrt{1^2 - 4(1)(-13)}}{2(1)} \][/tex]
[tex]\[ x = \frac{-1 \pm \sqrt{1 + 52}}{2} \][/tex]
[tex]\[ x = \frac{-1 \pm \sqrt{53}}{2} \][/tex]
The solutions are:
[tex]\[ x = \frac{-1 + \sqrt{53}}{2} \quad \text{and} \quad x = \frac{-1 - \sqrt{53}}{2} \][/tex]
So, the solutions are:
[tex]\[ x = \frac{-1 + \sqrt{53}}{2} \quad \text{and} \quad x = \frac{-1 - \sqrt{53}}{2} \][/tex]
### 2. Finding the [tex]\( y \)[/tex]-coordinates of the [tex]\( y \)[/tex]-intercepts
For [tex]\( y = x^2 - 2x - 4 \)[/tex], set [tex]\( x = 0 \)[/tex]:
[tex]\[ y = 0^2 - 2(0) - 4 = -4 \][/tex]
For [tex]\( y = -3x + 9 \)[/tex], set [tex]\( x = 0 \)[/tex]:
[tex]\[ y = -3(0) + 9 = 9 \][/tex]
So, the [tex]\( y \)[/tex]-coordinates of the y-intercepts are:
[tex]\[ y = -4 \quad \text{and} \quad y = 9 \][/tex]
### 3. Finding the [tex]\( x \)[/tex]-coordinates of the [tex]\( x \)[/tex]-intercepts
For the quadratic [tex]\( y = x^2 - 2x - 4 \)[/tex], set [tex]\( y = 0 \)[/tex]:
[tex]\[ x^2 - 2x - 4 = 0 \][/tex]
Solve for [tex]\( x \)[/tex] using the quadratic formula:
[tex]\[ x = \frac{2 \pm \sqrt{(-2)^2 - 4(1)(-4)}}{2(1)} \][/tex]
[tex]\[ x = \frac{2 \pm \sqrt{4 + 16}}{2} \][/tex]
[tex]\[ x = \frac{2 \pm \sqrt{20}}{2} \][/tex]
[tex]\[ x = \frac{2 \pm 2\sqrt{5}}{2} \][/tex]
[tex]\[ x = 1 \pm \sqrt{5} \][/tex]
So, the [tex]\( x \)[/tex]-coordinates of the x-intercepts are:
[tex]\[ x = 1 + \sqrt{5} \quad \text{and} \quad x = 1 - \sqrt{5} \][/tex]
For the linear equation [tex]\( y = -3x + 9 \)[/tex], set [tex]\( y = 0 \)[/tex]:
[tex]\[ -3x + 9 = 0 \][/tex]
Solve for [tex]\( x \)[/tex]:
[tex]\[ -3x = -9 \][/tex]
[tex]\[ x = 3 \][/tex]
So, the [tex]\( x \)[/tex]-coordinate of the x-intercept is:
[tex]\[ x = 3 \][/tex]
### 4. Finding the [tex]\( y \)[/tex]-coordinates of the Intersection Points
To find the [tex]\( y \)[/tex]-coordinates of the intersection points, we already have the [tex]\( x \)[/tex]-coordinates of the intersection points from solving [tex]\( x = \frac{-1 + \sqrt{53}}{2} \)[/tex] and [tex]\( x = \frac{-1 - \sqrt{53}}{2} \)[/tex]:
Substitute these [tex]\( x \)[/tex]-values back into either equation, such as [tex]\( y = x^2 - 2x - 4 \)[/tex].
For [tex]\( x = \frac{-1 + \sqrt{53}}{2} \)[/tex]:
[tex]\[ y = \left( \frac{-1 + \sqrt{53}}{2} \right)^2 - 2\left( \frac{-1 + \sqrt{53}}{2} \right) - 4 \][/tex]
[tex]\[ y \approx -0.420164833920778 \][/tex]
For [tex]\( x = \frac{-1 - \sqrt{53}}{2} \)[/tex]:
[tex]\[ y = \left( \frac{-1 - \sqrt{53}}{2} \right)^2 - 2\left( \frac{-1 - \sqrt{53}}{2} \right) - 4 \][/tex]
[tex]\[ y \approx 21.4201648339208 \][/tex]
So, the approximate [tex]\( y \)[/tex]-coordinates of the intersection points are:
[tex]\[ y \approx -0.420164833920778 \quad \text{and} \quad y \approx 21.4201648339208 \][/tex]
### 5. Finding the [tex]\( x \)[/tex]-coordinates of the Intersection Points
The [tex]\( x \)[/tex]-coordinates of the intersection points are:
[tex]\[ x = \frac{-1 + \sqrt{53}}{2} \quad \text{and} \quad x = \frac{-1 - \sqrt{53}}{2} \][/tex]
In summary:
- The solutions [tex]\( x \)[/tex]-coordinates: [tex]\( x = \frac{-1 + \sqrt{53}}{2} \quad \text{and} \quad x = \frac{-1 - \sqrt{53}}{2} \)[/tex].
- The [tex]\( y \)[/tex]-coordinates of the [tex]\( y \)[/tex]-intercepts are [tex]\( -4 \)[/tex] and [tex]\( 9 \)[/tex].
- The [tex]\( x \)[/tex]-coordinates of the [tex]\( x \)[/tex]-intercepts are [tex]\( 1 + \sqrt{5} \)[/tex], [tex]\( 1 - \sqrt{5} \)[/tex], and [tex]\( 3 \)[/tex].
- The [tex]\( y \)[/tex]-coordinates of the intersection points are approximately [tex]\( -0.420164833920778 \)[/tex] and [tex]\( 21.4201648339208 \)[/tex].
- The [tex]\( x \)[/tex]-coordinates of the intersection points are [tex]\( \frac{-1 + \sqrt{53}}{2} \)[/tex] and [tex]\( \frac{-1 - \sqrt{53}}{2} \)[/tex].
