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It seems like you are working on question 5 from the image. The problem asks to:

**Show algebraically that the line \( y = kx + 5 \) intersects the parabola \( y = x^2 + 2 \) twice for all values of \( k \).**

Let's go through the solution step-by-step.

### 1. Setting the two equations equal:
We need to find where the line \( y = kx + 5 \) intersects the parabola \( y = x^2 + 2 \). So we set the two equations equal to each other:

\[
kx + 5 = x^2 + 2
\]

### 2. Rearranging the equation:
Rearrange this to form a quadratic equation:

\[
x^2 - kx + 2 - 5 = 0
\]
\[
x^2 - kx - 3 = 0
\]

### 3. Solving the quadratic:
The quadratic equation is now \( x^2 - kx - 3 = 0 \). The discriminant of a quadratic equation \( ax^2 + bx + c = 0 \) is given by \( \Delta = b^2 - 4ac \). For the quadratic to have two real solutions, the discriminant must be positive or zero (\( \Delta \geq 0 \)).

Here, \( a = 1 \), \( b = -k \), and \( c = -3 \). So, the discriminant is:

\[
\Delta = (-k)^2 - 4(1)(-3)
\]
\[
\Delta = k^2 + 12
\]

### 4. Analyzing the discriminant:
Since \( \Delta = k^2 + 12 \), this value is always greater than zero for all values of \( k \) (since \( k^2 \geq 0 \) and 12 is positive). This means that the quadratic equation has two distinct real solutions for any value of \( k \).

### Conclusion:
The line \( y = kx + 5 \) will always intersect the parabola \( y = x^2 + 2 \) twice, regardless of the value of \( k \).

Would you like any further clarification or details on this? Here are some related questions you might want to explore:

1. Can the line ever be tangent to the parabola?
2. What happens to the intersection points when \( k = 0 \)?
3. Can we solve for the exact intersection points for a specific \( k \)?
4. How does the slope \( k \) affect the steepness of the line and its intersections?
5. Could we solve this problem using graphical methods?

**Tip:** Always analyze the discriminant of a quadratic equation to determine the number of solutions.

Show algebraically that the line y = kx + 5 intersects the parabola y = x^2 + 2 twice for all values of k.

Learn how to algebraically show that the line y = kx + 5 intersects the parabola y = x^2 + 2 twice for all values of k. This solution covers the quadratic equation and discriminant analysis, suitable for high school students in Grades 10-12.

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