\( (2 x+7)^{2}+(4 x-7)^{2}=4 x^{2} \)

Step 1 ：\( \text{Expand the squares}\: (2x+7)^{2} = 4x^{2} + 28x + 49 \)

Step 2 ：\( \text{Expand the squares}\: (4x-7)^{2} = 16x^{2} - 56x + 49 \)

Step 3 ：\( \text{Substitute into the equation}\: 4x^{2} + 28x + 49 + 16x^{2} - 56x + 49 = 4x^{2} \)

Step 4 ：\( \text{Simplify the equation}\: 16x^{2} - 28x + 49 = 0 \)

Step 5 ：\( \text{Solve for x}\: x \approx -1.0000,\: x\approx 1.9286 \)