Solve the following quadratic equation for x: 4x² + 4bx − (a² − b²) = 0

We know that for any quadratic equation of the form ax² + bx + c = 0
x = −b ± √b² − 4ac
2a
Given equation is 4x² + 4bx − a² + b² = 0
Hence, substituting the values of a, b and c in the above equation we get,
x = −4b ± √(4b)² − 4 × 4 × (−a² + b²)
2 × 4
= −4b ± √16b² + 16a² − 16b²
8
= −4b ± 4a
8
= −b ± a
2
So, x = −b + a
2 or x = −b − a
2
= −b + a
2
Hence, these are the values of x.