If the sum of the squares of two consecutive positive multiples of 4 is 656, then what is the mean of the two multiples?

Let the consecutive positive multiples of 4 be 4x and 4x + 4.

It is given that the sum of the squares of the two multiples is 656.

∴(4x)² + (4x + 4)² = 656

⇒ 16x² + 16x² + 16 + 32x = 656

⇒ 32x² + 32x + 16 = 656

⇒ 32x² + 32 x − 640 = 0

⇒ 32(x² + x − 20) = 0

⇒ x² + x − 20 = 0

⇒ x² + 5x − 4x − 20 = 0

⇒ x (x + 5) − 4 (x + 5) = 0

⇒ (x + 5) (x − 4) = 0

⇒ x + 5 = 0 or x − 4 = 0

⇒ x = −5 or x = 4

Since the multiples are positive, x ≠ −5.

∴ x = 4

Therefore, the multiples of 4 are 4x = 4 × 4 = 16 and 4x + 4 = 4 × 4 + 4 = 20

Thus, the required mean $\frac{16+20}{2}=18$.