Quadratic Equation:
The given quadratic equation is x^2 - 7(3 - 2k) - 2x(1 - 3k) = 0.

Equal Roots:
For a quadratic equation to have equal roots, the discriminant (b^2 - 4ac) should be equal to zero. In other words, the discriminant determines whether the roots of a quadratic equation are real and distinct (positive discriminant), real and equal (zero discriminant), or complex conjugates (negative discriminant).

Finding the Discriminant:
To find the discriminant, we need to determine the coefficients a, b, and c of the quadratic equation. Let's expand and simplify the given equation:

x^2 - 21 + 14k - 2x + 6kx = 0

Rearranging the terms, we get:

x^2 - 2x + 6kx - 21 + 14k = 0

Comparing this equation with the standard form ax^2 + bx + c = 0, we can determine the values of a, b, and c:

a = 1 (coefficient of x^2)
b = -2 + 6k (coefficient of x)
c = -21 + 14k (constant term)

Calculating the Discriminant:
The discriminant can be calculated using the formula: discriminant = b^2 - 4ac.

Substituting the values of a, b, and c into the formula, we have:

discriminant = (-2 + 6k)^2 - 4(1)(-21 + 14k)

Expanding and simplifying, we get:

discriminant = 4 - 24k + 36k^2 - 4(-21 + 14k)

discriminant = 4 - 24k + 36k^2 + 84 - 56k

discriminant = 88 - 80k + 36k^2

Equal Roots Condition:
For the quadratic equation to have equal roots, the discriminant should be equal to zero. Therefore, we set the discriminant equal to zero and solve for k:

88 - 80k + 36k^2 = 0

Simplifying, we get:

36k^2 - 80k + 88 = 0

Solving the Quadratic Equation:
To solve the quadratic equation, we can use the quadratic formula:

k = (-b ± √(b^2 - 4ac)) / (2a)

Substituting the values of a, b, and c into the quadratic formula, we have:

k = (-(-80) ± √((-80)^2 - 4(36)(88))) / (2(36))

k = (80 ± √(6400 - 12672)) / 72

k = (80 ± √(-6272)) / 72

The discriminant (-6272) is negative, indicating that the roots are complex conjugates. Therefore, there is no real value of k that will yield equal roots for the given quadratic equation.