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The given equation is:

\[
(3k - 4)x - 2k^2x^2 - 2 = 0
\]

It is stated that the equation has no real roots, and we are required to find the range of values for \( k \) that satisfy this condition.

### Step-by-Step Solution:

We can rewrite the given equation as a standard quadratic equation in terms of \( x \):

\[
-2k^2x^2 + (3k - 4)x - 2 = 0
\]

This is of the form \( ax^2 + bx + c = 0 \), where:
- \( a = -2k^2 \)
- \( b = 3k - 4 \)
- \( c = -2 \)

For a quadratic equation to have no real roots, its discriminant must be less than zero. The discriminant \( \Delta \) for a quadratic equation \( ax^2 + bx + c = 0 \) is given by:

\[
\Delta = b^2 - 4ac
\]

Substitute the values of \( a \), \( b \), and \( c \):

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

Simplify the expression:

\[
\Delta = (3k - 4)^2 - 16k^2
\]

Now expand both terms:

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

\[
\Delta = 9k^2 - 24k + 16 - 16k^2
\]

\[
\Delta = -7k^2 - 24k + 16
\]

For the quadratic to have no real roots, the discriminant must be less than zero:

\[
-7k^2 - 24k + 16 < 0
\]

### Solving the inequality:

We now solve the inequality \( -7k^2 - 24k + 16 < 0 \) by finding its critical points (roots). We do this by solving the equality \( -7k^2 - 24k + 16 = 0 \).

Use the quadratic formula:

\[
k = \frac{-(-24) \pm \sqrt{(-24)^2 - 4(-7)(16)}}{2(-7)}
\]

Simplify:

\[
k = \frac{24 \pm \sqrt{576 + 448}}{-14}
\]

\[
k = \frac{24 \pm \sqrt{1024}}{-14}
\]

\[
k = \frac{24 \pm 32}{-14}
\]

So, the two roots are:

\[
k = \frac{24 + 32}{-14} = \frac{56}{-14} = -4
\]
and
\[
k = \frac{24 - 32}{-14} = \frac{-8}{-14} = \frac{4}{7}
\]

Thus, the critical points are \( k = -4 \) and \( k = \frac{4}{7} \).

### Testing the intervals:

Since the quadratic \( -7k^2 - 24k + 16 \) opens downwards (the coefficient of \( k^2 \) is negative), the inequality will be satisfied between the roots. Therefore, the solution to the inequality is:

\[
-\frac{4}{7} < k < 4
\]

This is the range of values of \( k \) such that the given quadratic equation has no real roots.

### Final Answer:
\[
-\frac{4}{7} < k < 4
\]

Would you like further details or have any questions? Here are five related questions for deeper understanding:

1. How would the solution change if the discriminant were greater than zero?
2. What does it mean for a quadratic equation to have complex roots?
3. Could you derive the discriminant condition from first principles?
4. How would you graph the quadratic expression in \( k \)?
5. What happens to the quadratic if \( k = -4 \) or \( k = \frac{4}{7} \)?

**Tip**: Always check the sign of the leading coefficient when determining the direction of a parabola (whether it opens upwards or downwards).

The equation (3k - 4)x - 2k^2x^2 - 2 = 0 has no real roots. Find the possible range of values of k.

Learn how to find the range of values for k such that the quadratic equation (3k - 4)x - 2k^2x^2 - 2 = 0 has no real roots. The solution covers key algebraic concepts, including discriminant analysis and quadratic equations. Suitable for students in Grades 9-12.

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