Let's solve the given problem.

The quadratic equation is:

$x^2 - 4x = k$

Rewriting it in standard form:

$x^2 - 4x - k = 0$

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

$D = b^2 - 4ac$

Here:

• $a = 1$,
• $b = -4$,
• $c = -k$.

Substitute into the formula for $D$:

$D = (-4)^2 - 4(1)(-k)$

Simplify:

$D = 16 + 4k$

For the equation to have no real roots, we require:

$D < 0$

Substitute $D = 16 + 4k$ into the inequality:

$16 + 4k < 0$

Solve for $k$:

$4k < -16$

$k < -4$

Thus, the range of $k$ for which the quadratic equation has no real roots is:

$k < -4$

Would you like further clarification or detailed examples?

Here are 5 related questions to deepen your understanding:

1. What happens to the roots of the equation if $k = -4$?
2. How do we determine the nature of roots when $D = 0$?
3. Solve the equation when $k = -5$. Does it have real roots?
4. What is the significance of the discriminant in quadratic equations?
5. How does the range of $k$ change if the equation is $x^2 - 4x + k = 0$?

Tip: Always use the discriminant to analyze the nature of roots (real, equal, or imaginary) in quadratic equations!