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.