Let's solve the given equation [tex]\(4 x^4 - 4 x^2 = 8\)[/tex] step by step.
1. Rewrite the equation:
Start by moving all terms to one side of the equation:
[tex]\[
4 x^4 - 4 x^2 - 8 = 0
\][/tex]
2. Substitute [tex]\(\displaystyle u = x^2\)[/tex]:
Let [tex]\(u = x^2\)[/tex]. This substitution simplifies the equation because [tex]\(x^4\)[/tex] becomes [tex]\(u^2\)[/tex]. The equation then transforms into:
[tex]\[
4u^2 - 4u - 8 = 0
\][/tex]
3. Solve the quadratic equation:
We now solve for [tex]\(u\)[/tex] by using the quadratic formula [tex]\(u = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\)[/tex], where [tex]\( a = 4 \)[/tex], [tex]\( b = -4 \)[/tex], and [tex]\( c = -8 \)[/tex].
[tex]\[
u = \frac{-(-4) \pm \sqrt{(-4)^2 - 4 \cdot 4 \cdot (-8)}}{2 \cdot 4}
\][/tex]
[tex]\[
u = \frac{4 \pm \sqrt{16 + 128}}{8}
\][/tex]
[tex]\[
u = \frac{4 \pm \sqrt{144}}{8}
\][/tex]
[tex]\[
u = \frac{4 \pm 12}{8}
\][/tex]
This results in two possible values for [tex]\(u\)[/tex]:
[tex]\[
u = \frac{4 + 12}{8} = \frac{16}{8} = 2
\][/tex]
[tex]\[
u = \frac{4 - 12}{8} = \frac{-8}{8} = -1
\][/tex]
4. Back-substitute [tex]\( \displaystyle u = x^2 \)[/tex]:
- For [tex]\(u = 2\)[/tex]:
[tex]\[
x^2 = 2
\][/tex]
Taking the square root of both sides:
[tex]\[
x = \pm \sqrt{2}
\][/tex]
- For [tex]\(u = -1\)[/tex]:
[tex]\[
x^2 = -1
\][/tex]
Taking the square root of both sides:
[tex]\[
x = \pm i
\][/tex]
5. Collect all solutions:
The solutions are:
[tex]\[
x = \pm \sqrt{2} \quad \text{and} \quad x = \pm i
\][/tex]
6. Compare with given choices:
Among the provided options:
- [tex]\(\pm i\)[/tex] matches.
- [tex]\(\pm \sqrt{2}\)[/tex] matches.
Thus, the correct solutions for the equation [tex]\(4 x^4 - 4 x^2 = 8\)[/tex] are indeed
[tex]\[
x = \pm \sqrt{2}
\][/tex]
and
[tex]\[
x = \pm i.
\][/tex]
