Let's analyze and solve the problems shown in the image you uploaded.

Left Side Problem:

The graph of the quadratic function $y = x^2 - 2x - 8$ intersects the graph of the linear function $y = 2x + 2$. You are asked to find the x-coordinates of their points of intersection.

To solve this, we set the quadratic equation equal to the linear equation: $x^2 - 2x - 8 = 2x + 2$ Subtract $2x + 2$ from both sides: $x^2 - 4x - 10 = 0$

Now, we solve the quadratic equation using the quadratic formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ Here, $a = 1$, $b = -4$, and $c = -10$: $x = \frac{-(-4) \pm \sqrt{(-4)^2 - 4(1)(-10)}}{2(1)}$ $x = \frac{4 \pm \sqrt{16 + 40}}{2}$ $x = \frac{4 \pm \sqrt{56}}{2}$ $x = \frac{4 \pm 2\sqrt{14}}{2}$ $x = 2 \pm \sqrt{14}$

So, the x-coordinates of the points of intersection are: $x = 2 + \sqrt{14} \quad \text{and} \quad x = 2 - \sqrt{14}$

The correct option appears to be the one closest to these solutions.

Right Side Problem:

The graph of the linear function $y = 3x + 6$ intersects the x-axis. The question asks for the x-intercept of this line.

To find the x-intercept, set $y = 0$ and solve for $x$: $0 = 3x + 6$ $3x = -6$ $x = -2$

Thus, the x-intercept is $x = -2$, which corresponds to the correct answer.

Summary:

• Left Side Problem: The x-coordinates of the intersection points are $2 + \sqrt{14}$ and $2 - \sqrt{14}$.
• Right Side Problem: The x-intercept is $x = -2$.

Do you want more details on these solutions, or do you have any questions? Here are some related questions for further practice:

1. How do you determine the points of intersection between a quadratic and a linear equation?
2. What are the steps to solve a quadratic equation using the quadratic formula?
3. What does it mean for a line to intersect the x-axis?
4. How do you graphically determine the x-intercepts of a function?
5. What is the significance of the discriminant in the quadratic formula?
6. How would you solve a quadratic equation by completing the square?
7. Can a quadratic function and a linear function have more than two points of intersection? Why or why not?
8. How do you find the y-intercept of a linear function?

Tip: When solving quadratic equations, always check if the discriminant (the part under the square root) is positive, zero, or negative. This will tell you whether the equation has two real solutions, one real solution, or no real solutions, respectively.