What are the real or imaginary solutions of the polynomial equation x⁴ - 52x² + 576?

Solution:

Given x⁴ - 52x² + 576

Let x² = m

m² - 52m + 576 = 0

We use quadratic formula to find the roots

m = -b ± √(b² - 4ac) / 2a

Here, a = 1, b = -52, c = 576

m = -(-52) ± √((-52)² -4(576)) / 2

m = 52 ± √(2704 - 2304) / 2

m = 52 ± √(400) /2

m = 52 ± 20 /2

m = 72/2, 32/2

m = 36, 16 = x²

x = ±6, ±4.

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What are the real or imaginary solutions of the polynomial equation x⁴ - 52x² + 576?

Summary:

The real or imaginary solutions of the polynomial equation x⁴ - 52x² + 576 are ±6, ±4 and zero respectively.