Final answer:
The equation y⁴ - 20y² + 64 = 0 is solved by recognizing it as a quadratic in terms of y² and factoring it to (y² - 16)(y² - 4) = 0, yielding solutions y = ±4 and y = ±2.
Step-by-step explanation:
The equation y⁴ - 20y² + 64 = 0 can be solved by recognizing it as a quadratic equation in terms of y². To apply the quadratic factoring method, first identify a variable substitution such as t = y². The equation then becomes t² - 20t + 64 = 0, which is a standard quadratic equation.
Factoring this quadratic equation, we get (t - 16)(t - 4) = 0, so t = 16 or t = 4. Substituting back for y, we get y² = 16 or y² = 4.
Thus, the solutions for y are y = ±4 or y = ±2. We have four possible solutions for y, which are y = 4, y = -4, y = 2, and y = -2.