We are given the equation:

$(3k - 4)x - 2k^2 x^2 - 2 = 0$

This is a quadratic equation in $x$. To find the range of values of $k$ such that the equation has no real roots, we don't need to use the quadratic formula explicitly, but we can analyze the discriminant.

For a quadratic equation in the form $ax^2 + bx + c = 0$, the discriminant is:

$\Delta = b^2 - 4ac$

The quadratic equation has no real roots if $\Delta < 0$.

Step 1: Identify $a$, $b$, and $c$

We rewrite the given equation in standard form:

$-2k^2 x^2 + (3k - 4)x - 2 = 0$

Here, $a = -2k^2$, $b = 3k - 4$, and $c = -2$.

Step 2: Compute the discriminant

The discriminant is given by:

$\Delta = b^2 - 4ac = (3k - 4)^2 - 4(-2k^2)(-2)$

First, simplify each term:

• $(3k - 4)^2 = 9k^2 - 24k + 16$
• $4(-2k^2)(-2) = 16k^2$

Thus, the discriminant becomes:

$\Delta = (9k^2 - 24k + 16) - 16k^2 = 9k^2 - 16k^2 - 24k + 16$

Simplifying:

$\Delta = -7k^2 - 24k + 16$

Step 3: Set $\Delta < 0$

To ensure no real roots, we set the discriminant less than 0:

$-7k^2 - 24k + 16 < 0$

Now, we solve this inequality for $k$.

Step 4: Solve the quadratic inequality

First, multiply the inequality by $-1$ (which reverses the inequality sign):

$7k^2 + 24k - 16 > 0$

Next, solve the corresponding quadratic equation:

$7k^2 + 24k - 16 = 0$

Using the discriminant method (this is for the $k$-values of the boundary points):

$\Delta_k = 24^2 - 4(7)(-16) = 576 + 448 = 1024$

So, the solutions for $k$ are:

$k = \frac{-24 \pm \sqrt{1024}}{2 \times 7} = \frac{-24 \pm 32}{14}$

This gives:

$k_1 = \frac{-24 + 32}{14} = \frac{8}{14} = \frac{4}{7}, \quad k_2 = \frac{-24 - 32}{14} = \frac{-56}{14} = -4$

Thus, the quadratic inequality has roots at $k = \frac{4}{7}$ and $k = -4$.

Step 5: Determine the intervals

Since the parabola opens upwards (because the coefficient of $k^2$ is positive), the solution to $7k^2 + 24k - 16 > 0$ lies outside the interval between the roots. Therefore, the inequality is satisfied for:

$k < -4 \quad \text{or} \quad k > \frac{4}{7}$

Final Answer:

The range of values of $k$ for which the equation has no real roots is:

$k < -4 \quad \text{or} \quad k > \frac{4}{7}$

Would you like a more detailed explanation of any step or further assistance? Here are some related questions to explore:

1. How can we solve quadratic inequalities algebraically?
2. What are the different conditions for quadratic equations having real, complex, or no solutions?
3. How do we determine the nature of roots without using the quadratic formula?
4. Can the discriminant method be applied to higher-degree polynomials?
5. How does multiplying an inequality by a negative number affect the inequality sign?

Tip: Always remember to reverse the inequality sign when multiplying both sides by a negative number in an inequality.