To solve the equation [tex]\(9 x^4 - 2 x^2 - 7 = 0\)[/tex] using [tex]\(u\)[/tex]-substitution, follow these detailed steps:
### Step 1: [tex]\(u\)[/tex]-Substitution
Let [tex]\(u = x^2\)[/tex]. This transforms the original equation into a quadratic equation in terms of [tex]\(u\)[/tex]:
[tex]\[
9u^2 - 2u - 7 = 0.
\][/tex]
### Step 2: Solve the Quadratic Equation
We can now solve the quadratic equation [tex]\(9u^2 - 2u - 7 = 0\)[/tex] using the quadratic formula [tex]\(u = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\)[/tex], where [tex]\(a = 9\)[/tex], [tex]\(b = -2\)[/tex], and [tex]\(c = -7\)[/tex]:
1. Calculate the discriminant:
[tex]\[
\Delta = b^2 - 4ac = (-2)^2 - 4(9)(-7) = 4 + 252 = 256.
\][/tex]
2. Using the discriminant, calculate the roots:
[tex]\[
u = \frac{-(-2) \pm \sqrt{256}}{2 \cdot 9} = \frac{2 \pm 16}{18}.
\][/tex]
3. Simplify the individual solutions:
[tex]\[
u_1 = \frac{2 + 16}{18} = \frac{18}{18} = 1,
\][/tex]
[tex]\[
u_2 = \frac{2 - 16}{18} = \frac{-14}{18} = -\frac{7}{9}.
\][/tex]
### Step 3: Re-substitute [tex]\(u = x^2\)[/tex]
Now we need to find the values of [tex]\(x\)[/tex], knowing that [tex]\(u = x^2\)[/tex]:
1. For [tex]\(u = 1\)[/tex]:
[tex]\[
x^2 = 1 \implies x = \pm 1.
\][/tex]
2. For [tex]\(u = -\frac{7}{9}\)[/tex] (Note: [tex]\(u\)[/tex] must be non-negative since [tex]\(x^2\)[/tex] cannot be negative in the real number system. Therefore, solutions in the real number domain are nonexistent here for this value, and we'll consider complex solutions):
[tex]\[
x^2 = -\frac{7}{9} \implies x = \pm i \sqrt{\frac{7}{9}} = \pm i \frac{\sqrt{7}}{3}.
\][/tex]
### Step 4: Provide the Solutions
The solutions to the equation [tex]\(9 x^4 - 2 x^2 - 7 = 0\)[/tex] are:
[tex]\[
x = \pm 1 \quad \text{and} \quad x = \pm i \frac{\sqrt{7}}{3}.
\][/tex]
Matching these with the given multiple-choice options, we see that the correct solution is:
\[
\boxed{x = \pm i \frac{\sqrt{7}}{3} \quad \text{and} \quad x = \pm 1}.
