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The problem involves finding the range of values for \( k \) such that a line and a parabola do not intersect.

### Equations:
- **Line**: \( y = 2x - k \)
- **Parabola**: \( y = 3x^2 + 2kx + 5 \)

To find the range of \( k \) where the line and the parabola do **not intersect**, we can set the two equations equal to each other and analyze the resulting quadratic equation for conditions that ensure no real solutions.

### Step 1: Set the two equations equal
\[
2x - k = 3x^2 + 2kx + 5
\]

Rearrange the equation to form a quadratic:
\[
0 = 3x^2 + (2k - 2)x + (5 + k)
\]
\[
3x^2 + (2k - 2)x + (5 + k) = 0
\]

### Step 2: Use the discriminant
For a quadratic equation \( ax^2 + bx + c = 0 \), the discriminant is:
\[
\Delta = b^2 - 4ac
\]
If \( \Delta < 0 \), the quadratic has no real solutions, meaning the line and the parabola do not intersect.

Here, \( a = 3 \), \( b = 2k - 2 \), and \( c = 5 + k \). The discriminant is:
\[
\Delta = (2k - 2)^2 - 4(3)(5 + k)
\]
Simplify:
\[
\Delta = (2k - 2)^2 - 12(5 + k)
\]
\[
\Delta = (4k^2 - 8k + 4) - (60 + 12k)
\]
\[
\Delta = 4k^2 - 8k + 4 - 60 - 12k
\]
\[
\Delta = 4k^2 - 20k - 56
\]

### Step 3: Set discriminant less than 0
For no intersection, we need \( \Delta < 0 \):
\[
4k^2 - 20k - 56 < 0
\]
Divide by 4:
\[
k^2 - 5k - 14 < 0
\]

### Step 4: Solve the inequality
Solve \( k^2 - 5k - 14 = 0 \) using the quadratic formula:
\[
k = \frac{-(-5) \pm \sqrt{(-5)^2 - 4(1)(-14)}}{2(1)}
\]
\[
k = \frac{5 \pm \sqrt{25 + 56}}{2}
\]
\[
k = \frac{5 \pm \sqrt{81}}{2}
\]
\[
k = \frac{5 \pm 9}{2}
\]
\[
k_1 = \frac{5 + 9}{2} = 7 \quad \text{and} \quad k_2 = \frac{5 - 9}{2} = -2
\]

The solutions to the equation are \( k = 7 \) and \( k = -2 \).

### Step 5: Analyze the inequality
The quadratic inequality \( k^2 - 5k - 14 < 0 \) holds between the roots, so the range of \( k \) for which the line and the parabola do not intersect is:
\[
-2 < k < 7
\]

### Final Answer:
The line and the parabola do not intersect for \( k \) in the range:
\[
-2 < k < 7
\]

---

Would you like further details or have any questions?

Here are 5 related questions to explore:
1. How do we determine if a quadratic has no real solutions using the discriminant?
2. What changes when the discriminant is exactly zero?
3. How can we graphically interpret the intersection of a line and a parabola?
4. How does the quadratic formula relate to the roots of an equation?
5. Can you apply this method to any line and parabola intersection problem?

**Tip:** Always check if a quadratic equation can be factored easily before using the quadratic formula! It can save time.

A straight line has equation y = 2x - k and a parabola has equation y = 3x^2 + 2kx + 5 where k is a constant. Find the range of values of k for which the line and the parabola do not intersect.

Explore how to find the range of k for which the line y = 2x - k and the parabola y = 3x^2 + 2kx + 5 do not intersect. Learn about quadratic equations, the discriminant, and inequalities with a detailed solution. Suitable for students in Grades 10-12.

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