What are the solutions of the equation x⁴ + 6x² + 5 = 0? Use u substitution to solve.

Solution: 

Given: Equation is x⁴ + 6x² + 5 = 0

Let x² = u then the given equation becomes u² + 6u + 5 = 0

This is in the form ax² + bx + c=0

We can solve by using quadratic formula for finding roots

x = {-b ± √(b² - 4ac)} / 2a

Here, a = 1, b = 6, c = 5

u = {-(6) ± √((6)² - 4(1)(5))} / 2(1)

u= {-6 ± √(36 -20)} / 2

u = {-6 ± √16}/2

u = {-6 ± 4} /2

u = -1, or u = -5

x² = u

x = ±√u

x = ±√-1 or x = ±√-5

x = ±i or x = ±√5i

The solutions are + i, -i , +√5i, -√5i for the given equation.

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What are the solutions of the equation x⁴ + 6x² + 5 = 0? Use u substitution to solve.

Summary:

The solutions are + i, -i , +√5i, -√5i of the equation x⁴ + 6x² + 5 = 0