To solve the equation [tex]\(x^4 - 9x^2 + 8 = 0\)[/tex] using [tex]\(u\)[/tex]-substitution, follow these steps:
1. Substitution:
First, we substitute [tex]\(u = x^2\)[/tex]. This transforms our equation in terms of [tex]\(u\)[/tex]:
[tex]\[
(x^2)^2 - 9(x^2) + 8 = 0 \implies u^2 - 9u + 8 = 0
\][/tex]
2. Solve the Quadratic Equation:
Now we solve the quadratic equation [tex]\(u^2 - 9u + 8 = 0\)[/tex]:
[tex]\[
u^2 - 9u + 8 = 0
\][/tex]
This can be factored or solved using the quadratic formula, [tex]\(u = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\)[/tex]:
Here, [tex]\(a = 1\)[/tex], [tex]\(b = -9\)[/tex], and [tex]\(c = 8\)[/tex].
[tex]\[
u = \frac{9 \pm \sqrt{81 - 32}}{2} = \frac{9 \pm \sqrt{49}}{2} = \frac{9 \pm 7}{2}
\][/tex]
This gives us two solutions for [tex]\(u\)[/tex]:
[tex]\[
u = \frac{9 + 7}{2} = 8 \quad \text{and} \quad u = \frac{9 - 7}{2} = 1
\][/tex]
So, the solutions for [tex]\(u\)[/tex] are [tex]\(u = 8\)[/tex] and [tex]\(u = 1\)[/tex].
3. Back-substitute [tex]\(u = x^2\)[/tex] and solve for [tex]\(x\)[/tex]:
Now we substitute back [tex]\(u = x^2\)[/tex]:
[tex]\[
x^2 = 8 \quad \Rightarrow \quad x = \pm \sqrt{8} = \pm 2\sqrt{2}
\][/tex]
and
[tex]\[
x^2 = 1 \quad \Rightarrow \quad x = \pm \sqrt{1} = \pm 1
\][/tex]
4. Combine the solutions:
Combining all these solutions, we get:
[tex]\[
x = \pm 1 \quad \text{and} \quad x = \pm 2\sqrt{2}
\][/tex]
Therefore, the complete set of solutions to the equation [tex]\(x^4 - 9x^2 + 8 = 0\)[/tex] is:
[tex]\[
x = \pm 1 \quad \text{and} \quad x = \pm 2\sqrt{2}
\][/tex]
So the correct answer is:
[tex]\[
x = \pm 1 \quad \text{and} \quad x = \pm 2 \sqrt{2}
\][/tex]
