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.