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

The answers are x = - 1, 1, i√7/3, - i√7/3.

Solution:

Solving by making a u-substitution, we let u = x² and rewrite the equation in quadratic form.

9u² - 2u - 7 = 0

We can now solve the quadratic equation by factoring. We need two numbers whose sum is - 2 and whose product is - 7. In this case, it would have to be 7 and - 1, considering the term 9u². Hence, we can also write our equation in the factored form

(u - 1) (9u + 7) = 0

Now we have a product of two expressions that is equal to zero, which means any u value that makes either (u - 1) or (9u + 7) zero will make their product zero.

u - 1 = 0 = > u = 1

9u + 7 = 0 = > u = - 7/9

We substitute back x² = u to calculate for x.

u = 1 = > x² = 1 = > x = - 1, 1

u = - 7/9 = > x² = - 7/9 = > x = i√7/3, - i√7/3

Therefore, the solutions are - 1, 1, i√7/3, and - i√7/3.