Westonci.ca is your trusted source for finding answers to all your questions. Ask, explore, and learn with our expert community. Get accurate and detailed answers to your questions from a dedicated community of experts on our Q&A platform. Join our Q&A platform to connect with experts dedicated to providing accurate answers to your questions in various fields.
Sagot :
Sure, let's go through each part of the problem step-by-step.
### 1. Solutions of [tex]\( x^2 - 2x - 4 = -3x + 9 \)[/tex]
To solve the equation [tex]\( x^2 - 2x - 4 = -3x + 9 \)[/tex], we start by setting the equations equal to each other and solving for [tex]\( x \)[/tex].
[tex]\[ x^2 - 2x - 4 = -3x + 9 \][/tex]
We bring all terms to one side to form a standard quadratic equation:
[tex]\[ x^2 - 2x - 4 + 3x - 9 = 0 \][/tex]
[tex]\[ x^2 + x - 13 = 0 \][/tex]
Solving this quadratic equation, we find the solutions:
[tex]\[ x = -\frac{1}{2} + \frac{\sqrt{53}}{2} \][/tex]
[tex]\[ x = -\frac{\sqrt{53}}{2} - \frac{1}{2} \][/tex]
Thus, the solutions for the intersection points are:
[tex]\[ x_1 = -\frac{1}{2} + \frac{\sqrt{53}}{2} \][/tex]
[tex]\[ x_2 = -\frac{\sqrt{53}}{2} - \frac{1}{2} \][/tex]
### 2. [tex]\( y \)[/tex]-Coordinates of the [tex]\( y \)[/tex]-Intercepts
The [tex]\( y \)[/tex]-intercept is found by setting [tex]\( x = 0 \)[/tex] in the equation:
For [tex]\( y = x^2 - 2x - 4 \)[/tex]:
[tex]\[ y = 0^2 - 2 \cdot 0 - 4 \][/tex]
[tex]\[ y = -4 \][/tex]
For [tex]\( y = -3x + 9 \)[/tex]:
[tex]\[ y = -3 \cdot 0 + 9 \][/tex]
[tex]\[ y = 9 \][/tex]
Thus, the [tex]\( y \)[/tex]-intercepts are:
[tex]\[ y_{\text{intercept of } y = x^2 - 2x - 4} = -4 \][/tex]
[tex]\[ y_{\text{intercept of } y = -3x + 9} = 9 \][/tex]
### 3. [tex]\( x \)[/tex]-Coordinates of the [tex]\( x \)[/tex]-Intercepts
The [tex]\( x \)[/tex]-intercept is found by setting [tex]\( y = 0 \)[/tex] in the equation:
For [tex]\( y = x^2 - 2x - 4 \)[/tex]:
[tex]\[ x^2 - 2x - 4 = 0 \][/tex]
Solving the above quadratic equation, we get:
[tex]\[ x = 1 - \sqrt{5} \][/tex]
[tex]\[ x = 1 + \sqrt{5} \][/tex]
For [tex]\( y = -3x + 9 \)[/tex]:
[tex]\[ 0 = -3x + 9 \][/tex]
[tex]\[ 3x = 9 \][/tex]
[tex]\[ x = 3 \][/tex]
Thus, the [tex]\( x \)[/tex]-intercepts are:
[tex]\[ x_{\text{intercepts of } y = x^2 - 2x - 4} = 1 - \sqrt{5}, \; 1 + \sqrt{5} \][/tex]
[tex]\[ x_{\text{intercept of } y = -3x + 9} = 3 \][/tex]
### 4. [tex]\( y \)[/tex]-Coordinates of the Intersection Points
We have already found the [tex]\( x \)[/tex]-coordinates of the intersection points as:
[tex]\[ x_1 = -\frac{1}{2} + \frac{\sqrt{53}}{2} \][/tex]
[tex]\[ x_2 = -\frac{\sqrt{53}}{2} - \frac{1}{2} \][/tex]
To find the [tex]\( y \)[/tex]-coordinates of these intersection points, we substitute these [tex]\( x \)[/tex]-values back into either of the original equations (since they are equal at the intersection points):
For [tex]\( x_1 = -\frac{1}{2} + \frac{\sqrt{53}}{2} \)[/tex]:
[tex]\[ y_1 = \left( -\frac{1}{2} + \frac{\sqrt{53}}{2} \right)^2 - 2 \left( -\frac{1}{2} + \frac{\sqrt{53}}{2} \right) - 4 \][/tex]
For [tex]\( x_2 = -\frac{\sqrt{53}}{2} - \frac{1}{2} \)[/tex]:
[tex]\[ y_2 = \left( -\frac{\sqrt{53}}{2} - \frac{1}{2} \right)^2 - 2 \left( -\frac{\sqrt{53}}{2} - \frac{1}{2} \right) - 4 \][/tex]
So, the [tex]\( y \)[/tex]-coordinates of the intersection points are:
[tex]\[ y_1 = -\sqrt{53} - 3 + \left( -\frac{1}{2} + \frac{\sqrt{53}}{2} \right)^2 \][/tex]
[tex]\[ y_2 = -3 + \sqrt{53} + \left( -\frac{\sqrt{53}}{2} - \frac{1}{2} \right)^2 \][/tex]
### 5. [tex]\( x \)[/tex]-Coordinates of the Intersection Points
The solutions are found in step 1:
[tex]\[ x_1 = -\frac{1}{2} + \frac{\sqrt{53}}{2} \][/tex]
[tex]\[ x_2 = -\frac{\sqrt{53}}{2} - \frac{1}{2} \][/tex]
Putting it all together, the results are as follows:
- [tex]\( x \)[/tex]-Coordinates of intersection points: [tex]\( x_1 = -\frac{1}{2} + \frac{\sqrt{53}}{2} \)[/tex], [tex]\( x_2 = -\frac{\sqrt{53}}{2} - \frac{1}{2} \)[/tex]
- [tex]\( y \)[/tex]-Intercepts: [tex]\( y_{\text{intercept of } y = x^2 - 2x - 4} = -4 \)[/tex], [tex]\( y_{\text{intercept of } y = -3x + 9} = 9 \)[/tex]
- [tex]\( x \)[/tex]-Intercepts: [tex]\( x_{\text{intercepts of } y = x^2 - 2x - 4} = 1 - \sqrt{5}, 1 + \sqrt{5} \)[/tex], [tex]\( x_{\text{intercept of } y = -3x + 9} = 3 \)[/tex]
- [tex]\( y \)[/tex]-Coordinates of intersection points: [tex]\( y_1 = -\sqrt{53} - 3 + \left( -\frac{1}{2} + \frac{\sqrt{53}}{2} \right)^2 \)[/tex], [tex]\( y_2 = -3 + \sqrt{53} + \left( -\frac{\sqrt{53}}{2} - \frac{1}{2} \right)^2 \)[/tex]
### 1. Solutions of [tex]\( x^2 - 2x - 4 = -3x + 9 \)[/tex]
To solve the equation [tex]\( x^2 - 2x - 4 = -3x + 9 \)[/tex], we start by setting the equations equal to each other and solving for [tex]\( x \)[/tex].
[tex]\[ x^2 - 2x - 4 = -3x + 9 \][/tex]
We bring all terms to one side to form a standard quadratic equation:
[tex]\[ x^2 - 2x - 4 + 3x - 9 = 0 \][/tex]
[tex]\[ x^2 + x - 13 = 0 \][/tex]
Solving this quadratic equation, we find the solutions:
[tex]\[ x = -\frac{1}{2} + \frac{\sqrt{53}}{2} \][/tex]
[tex]\[ x = -\frac{\sqrt{53}}{2} - \frac{1}{2} \][/tex]
Thus, the solutions for the intersection points are:
[tex]\[ x_1 = -\frac{1}{2} + \frac{\sqrt{53}}{2} \][/tex]
[tex]\[ x_2 = -\frac{\sqrt{53}}{2} - \frac{1}{2} \][/tex]
### 2. [tex]\( y \)[/tex]-Coordinates of the [tex]\( y \)[/tex]-Intercepts
The [tex]\( y \)[/tex]-intercept is found by setting [tex]\( x = 0 \)[/tex] in the equation:
For [tex]\( y = x^2 - 2x - 4 \)[/tex]:
[tex]\[ y = 0^2 - 2 \cdot 0 - 4 \][/tex]
[tex]\[ y = -4 \][/tex]
For [tex]\( y = -3x + 9 \)[/tex]:
[tex]\[ y = -3 \cdot 0 + 9 \][/tex]
[tex]\[ y = 9 \][/tex]
Thus, the [tex]\( y \)[/tex]-intercepts are:
[tex]\[ y_{\text{intercept of } y = x^2 - 2x - 4} = -4 \][/tex]
[tex]\[ y_{\text{intercept of } y = -3x + 9} = 9 \][/tex]
### 3. [tex]\( x \)[/tex]-Coordinates of the [tex]\( x \)[/tex]-Intercepts
The [tex]\( x \)[/tex]-intercept is found by setting [tex]\( y = 0 \)[/tex] in the equation:
For [tex]\( y = x^2 - 2x - 4 \)[/tex]:
[tex]\[ x^2 - 2x - 4 = 0 \][/tex]
Solving the above quadratic equation, we get:
[tex]\[ x = 1 - \sqrt{5} \][/tex]
[tex]\[ x = 1 + \sqrt{5} \][/tex]
For [tex]\( y = -3x + 9 \)[/tex]:
[tex]\[ 0 = -3x + 9 \][/tex]
[tex]\[ 3x = 9 \][/tex]
[tex]\[ x = 3 \][/tex]
Thus, the [tex]\( x \)[/tex]-intercepts are:
[tex]\[ x_{\text{intercepts of } y = x^2 - 2x - 4} = 1 - \sqrt{5}, \; 1 + \sqrt{5} \][/tex]
[tex]\[ x_{\text{intercept of } y = -3x + 9} = 3 \][/tex]
### 4. [tex]\( y \)[/tex]-Coordinates of the Intersection Points
We have already found the [tex]\( x \)[/tex]-coordinates of the intersection points as:
[tex]\[ x_1 = -\frac{1}{2} + \frac{\sqrt{53}}{2} \][/tex]
[tex]\[ x_2 = -\frac{\sqrt{53}}{2} - \frac{1}{2} \][/tex]
To find the [tex]\( y \)[/tex]-coordinates of these intersection points, we substitute these [tex]\( x \)[/tex]-values back into either of the original equations (since they are equal at the intersection points):
For [tex]\( x_1 = -\frac{1}{2} + \frac{\sqrt{53}}{2} \)[/tex]:
[tex]\[ y_1 = \left( -\frac{1}{2} + \frac{\sqrt{53}}{2} \right)^2 - 2 \left( -\frac{1}{2} + \frac{\sqrt{53}}{2} \right) - 4 \][/tex]
For [tex]\( x_2 = -\frac{\sqrt{53}}{2} - \frac{1}{2} \)[/tex]:
[tex]\[ y_2 = \left( -\frac{\sqrt{53}}{2} - \frac{1}{2} \right)^2 - 2 \left( -\frac{\sqrt{53}}{2} - \frac{1}{2} \right) - 4 \][/tex]
So, the [tex]\( y \)[/tex]-coordinates of the intersection points are:
[tex]\[ y_1 = -\sqrt{53} - 3 + \left( -\frac{1}{2} + \frac{\sqrt{53}}{2} \right)^2 \][/tex]
[tex]\[ y_2 = -3 + \sqrt{53} + \left( -\frac{\sqrt{53}}{2} - \frac{1}{2} \right)^2 \][/tex]
### 5. [tex]\( x \)[/tex]-Coordinates of the Intersection Points
The solutions are found in step 1:
[tex]\[ x_1 = -\frac{1}{2} + \frac{\sqrt{53}}{2} \][/tex]
[tex]\[ x_2 = -\frac{\sqrt{53}}{2} - \frac{1}{2} \][/tex]
Putting it all together, the results are as follows:
- [tex]\( x \)[/tex]-Coordinates of intersection points: [tex]\( x_1 = -\frac{1}{2} + \frac{\sqrt{53}}{2} \)[/tex], [tex]\( x_2 = -\frac{\sqrt{53}}{2} - \frac{1}{2} \)[/tex]
- [tex]\( y \)[/tex]-Intercepts: [tex]\( y_{\text{intercept of } y = x^2 - 2x - 4} = -4 \)[/tex], [tex]\( y_{\text{intercept of } y = -3x + 9} = 9 \)[/tex]
- [tex]\( x \)[/tex]-Intercepts: [tex]\( x_{\text{intercepts of } y = x^2 - 2x - 4} = 1 - \sqrt{5}, 1 + \sqrt{5} \)[/tex], [tex]\( x_{\text{intercept of } y = -3x + 9} = 3 \)[/tex]
- [tex]\( y \)[/tex]-Coordinates of intersection points: [tex]\( y_1 = -\sqrt{53} - 3 + \left( -\frac{1}{2} + \frac{\sqrt{53}}{2} \right)^2 \)[/tex], [tex]\( y_2 = -3 + \sqrt{53} + \left( -\frac{\sqrt{53}}{2} - \frac{1}{2} \right)^2 \)[/tex]
Thanks for using our service. We're always here to provide accurate and up-to-date answers to all your queries. Thank you for your visit. We're dedicated to helping you find the information you need, whenever you need it. Thank you for visiting Westonci.ca. Stay informed by coming back for more detailed answers.
