1. factor the expression over the complex numbers. x^2+24 2. what number needs to be added to both sides of the equation to complete the square? x^2−9x=8 enter your answers in the boxes. (picture included) 3. factor the expression completely over the complex numbers. y^4+40y^2+400

1. (x² + 24) factors as the difference of two squares (x² - (-24)) where -24 is the square of the complex number i√24 = 2i√6. The factoring is
  (x - 2i√6)(x + 2i√6)

2. The square of half the coefficient of the x-term needs to be added: 4.5² = 20.25
This number goes in both boxes.

3. This is the square of (y² +20), so the factorization is similar to that for problem 1.
  y⁴ + 40y² + 400 = (y² + 20)² = (y -2i√5)²(y +2i√5)²

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