This problem involves finding the value of a coefficient in a quadratic equation based on the nature of its roots. The key is understanding the condition for real and equal roots.

Quadratic Equation Analysis

The given quadratic equation is:

$$kx^2 – 4x + 1 = 0$$

For a general quadratic equation of the form $ax^2 + bx + c = 0$, we can identify the coefficients:

• $a = k$
• $b = -4$
• $c = 1$

Condition for Real and Equal Roots

A quadratic equation has real and equal roots when its discriminant is equal to zero. The discriminant ($D$) is calculated using the formula:

$$D = b^2 - 4ac$$

The condition for real and equal roots is:

$$D = 0$$

Solving for the Value of k

We substitute the coefficients ($a=k$, $b=-4$, $c=1$) into the discriminant formula and set it to zero:

1. Substitute coefficients:

   $$(-4)^2 - 4(k)(1) = 0$$

2. Simplify the equation:

   $$16 - 4k = 0$$

3. Isolate the term with k:

   $$16 = 4k$$

4. Solve for k:

   Divide both sides by 4:

   $$k = \frac{16}{4}$$

   $$k = 4$$

Therefore, the value of k that makes the roots of the quadratic equation $kx^2 – 4x + 1 = 0$ real and equal is 4.