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).