To find the solutions of the equation [tex]\(x^4 - 9x^2 + 8 = 0\)[/tex], let's use the substitution method where we let [tex]\(u = x^2\)[/tex]. This simplifies our equation into a quadratic form, as follows:
1. Substitute [tex]\(u = x^2\)[/tex]:
The equation transforms to:
[tex]\[
u^2 - 9u + 8 = 0
\][/tex]
2. Solve the quadratic equation:
The equation [tex]\(u^2 - 9u + 8 = 0\)[/tex] can be 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]. Plugging in these values, we get:
[tex]\[
u = \frac{-(-9) \pm \sqrt{(-9)^2 - 4 \cdot 1 \cdot 8}}{2 \cdot 1}
\][/tex]
[tex]\[
u = \frac{9 \pm \sqrt{81 - 32}}{2}
\][/tex]
[tex]\[
u = \frac{9 \pm \sqrt{49}}{2}
\][/tex]
[tex]\[
u = \frac{9 \pm 7}{2}
\][/tex]
Therefore, we have two solutions for [tex]\(u\)[/tex]:
[tex]\[
u = \frac{9 + 7}{2} = \frac{16}{2} = 8
\][/tex]
[tex]\[
u = \frac{9 - 7}{2} = \frac{2}{2} = 1
\][/tex]
3. Back-substitute [tex]\(u = x^2\)[/tex]:
We now have two equations in terms of [tex]\(x\)[/tex]:
[tex]\[
x^2 = 8
\][/tex]
[tex]\[
x^2 = 1
\][/tex]
4. Solve for [tex]\(x\)[/tex]:
For [tex]\(x^2 = 8\)[/tex]:
[tex]\[
x = \pm \sqrt{8}
\][/tex]
Simplifying, we get:
[tex]\[
x = \pm 2\sqrt{2}
\][/tex]
For [tex]\(x^2 = 1\)[/tex]:
[tex]\[
x = \pm \sqrt{1}
\][/tex]
Simplifying, we get:
[tex]\[
x = \pm 1
\][/tex]
5. Combine all solutions:
Therefore, the complete set of solutions for the original equation [tex]\(x^4 - 9x^2 + 8 = 0\)[/tex] is:
[tex]\[
x = \pm 1, \pm 2\sqrt{2}
\][/tex]
This matches the option:
[tex]\[
x = \pm 1 \text{ and } x = \pm 2\sqrt{2}
\][/tex]
