How can I solve the equation x^4 + 6x^2 + 5 = 0 using u-substitution?

To solve the equation x⁴ + 6x² + 5 = 0 using u-substitution, we first make a substitution that simplifies the equation.

Let u = x². Then, we can express x⁴ as u², leading to the transformation of the equation:

u² + 6u + 5 = 0

This quadratic equation can now be solved using the quadratic formula, u = (-b ± √(b² – 4ac)) / 2a, where:

• a = 1
• b = 6
• c = 5

Substituting the values into the formula gives us:

u = (-6 ± √(6² – 4 * 1 * 5)) / (2 * 1)

Calculating the discriminant:

• 6² = 36
• 4 * 1 * 5 = 20
• 36 – 20 = 16

Now substituting back:

u = (-6 ± √16) / 2

This simplifies to:

u = (-6 ± 4) / 2

Thus, we have two potential solutions for u:

• u = (-6 + 4) / 2 = -2 / 2 = -1
• u = (-6 – 4) / 2 = -10 / 2 = -5

Recall that we made the substitution u = x². Therefore, we now have:

• x² = -1
• x² = -5

Since both equations produce negative values, the solutions will involve complex numbers:

• For x² = -1, we get x = i and x = -i.
• For x² = -5, we find x = √5i and x = -√5i.

Thus, the complete set of solutions to the original equation x⁴ + 6x² + 5 = 0 is:

• x = i
• x = -i
• x = √5i
• x = -√5i