Question

Factor the expression over the complex numbers. x^2+50

Answer 1

Note that
i² = -1 by definition.

Also,
a² - b² = (a+b)(a-b)

Consider the expression
x² - [i√(50)]² = (x + i√50)(x - i√50)
Also,
x² - (i√50)² = x² - i²(√50)²
                   = x² - (-1)*50
                   = x² + 50

Note that
√50 = √[(25*2] = √25 * √2 = 5√2.

Therefore
x² + 50 = (x + 5√2 i)(x - 5√2 i)

Answer: $(x+5 \sqrt{2} i)(x-5 \sqrt{2} i)$