To solve the equation \( (13x - 7)^2 = 4(3x - 4)^2 \), follow these steps:

1. **Expand Both Sides**:
- Left Side: \( (13x - 7)^2 = 169x^2 - 182x + 49 \)
- Right Side: \( 4(3x - 4)^2 = 4(9x^2 - 24x + 16) = 36x^2 - 96x + 64 \)

2. **Set the Equation to Zero**:
\[
169x^2 - 182x + 49 = 36x^2 - 96x + 64
\]
Rearranging gives:
\[
169x^2 - 36x^2 - 182x + 96x + 49 - 64 = 0
\]
Simplifying that results in:
\[
133x^2 - 86x - 15 = 0
\]

3. **Use the Quadratic Formula**:
The quadratic formula is given by:
\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]
Here, \( a = 133 \), \( b = -86 \), and \( c = -15 \).

- Calculate the discriminant:
\[
b^2 - 4ac = (-86)^2 - 4(133)(-15) = 7396 + 7980 = 15376
\]

- Now, find \( x \):
\[
x = \frac{86 \pm \sqrt{15376}}{2 \cdot 133}
\]

Simplifying \(\sqrt{15376}\):
\[
\sqrt{15376} = 124
\]
So,
\[
x = \frac{86 \pm 124}{266}
\]

- The two potential solutions for \( x \) are:
\[
x_1 = \frac{210}{266} = \frac{105}{133}, \quad x_2 = \frac{-38}{266} = \frac{-19}{133}
\]

4. **Final Solutions**:
The solutions to the equation \( (13x - 7)^2 = 4(3x - 4)^2 \) are:
\[
x = \frac{105}{133}, \quad x = \frac{-19}{133}
\]