What are the solutions of the equation x^4 – 6x^2 + 5 = 0? Use u substitution to solve.

To solve the equation x⁴ – 6x² + 5 = 0, we can use a substitution method. Let’s start by making a substitution to simplify the equation.

Let u = x². Then, the equation can be rewritten in terms of u:

u² – 6u + 5 = 0

Now, we need to factor this quadratic equation. We look for two numbers that multiply to +5 and add up to -6. These numbers are -5 and -1. Thus, we can factor the equation as:

(u – 5)(u – 1) = 0

Setting each factor equal to zero, we have:

• u – 5 = 0 ⟹ u = 5
• u – 1 = 0 ⟹ u = 1

Now we substitute back u = x² into the equations:

• From u = 5:
  x² = 5 ⟹ x = ±√5
• From u = 1:
  x² = 1 ⟹ x = ±1

Thus, the solutions to the original equation x⁴ – 6x² + 5 = 0 are:

• x = √5
• x = -√5
• x = 1
• x = -1

In conclusion, the complete set of solutions is ±√5 and ±1.