What are the solutions of the equation 9x^4 + 2x^2 + 7 = 0? Use u substitution to solve.

To solve the equation 9x⁴ + 2x² + 7 = 0 using u substitution, we first let u = x². This gives us:

9u² + 2u + 7 = 0

This is a quadratic equation in terms of u. We can use the quadratic formula, u = (-b ± √(b² – 4ac)) / 2a, where a = 9, b = 2, and c = 7.

Calculating the discriminant:

b² – 4ac = 2² – 4(9)(7) = 4 – 252 = -248

Since the discriminant is negative, there are no real solutions for u. However, we can find complex solutions:

u = (-2 ± √(-248)) / (2 * 9)

Calculation of the square root of -248 gives us:

√(-248) = √248 * i = √(4 * 62) * i = 2√62 * i

Plugging this back into our equation for u:

u = (-2 ± 2√62 * i) / 18 = -1/9 ± √62 * i / 9

Now, recall that we let u = x². Thus, we have:

x² = -1/9 ± √62 * i / 9

To find x, we take the square root of both sides:

x = ±√(-1/9 ± √62 * i / 9)

This means that there are complex solutions for x, and they can be expressed as:

x = ±(1/3)√(1 ± √62 * i)

Thus, the solutions to the original equation in terms of complex numbers are:

x = ±(1/3)√(1 + √62 * i) and x = ±(1/3)√(1 – √62 * i).