To solve the equation [tex]\( x^4 - 5x^2 - 14 = 0 \)[/tex] using factoring, let's proceed with the following steps:
1. Introduce a substitution to simplify the equation. Let [tex]\( y = x^2 \)[/tex]. Therefore, the equation can be rewritten in terms of [tex]\( y \)[/tex]:
[tex]\[
y^2 - 5y - 14 = 0
\][/tex]
2. Recognize this as a quadratic equation in [tex]\( y \)[/tex]. We can solve it using the quadratic formula [tex]\( y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)[/tex], where [tex]\( a = 1 \)[/tex], [tex]\( b = -5 \)[/tex], and [tex]\( c = -14 \)[/tex].
3. Apply the quadratic formula:
[tex]\[
y = \frac{-(-5) \pm \sqrt{(-5)^2 - 4 \cdot 1 \cdot (-14)}}{2 \cdot 1} = \frac{5 \pm \sqrt{25 + 56}}{2} = \frac{5 \pm \sqrt{81}}{2} = \frac{5 \pm 9}{2}
\][/tex]
4. Solve for [tex]\( y \)[/tex]:
[tex]\[
y = \frac{5 + 9}{2} = 7 \quad \text{and} \quad y = \frac{5 - 9}{2} = -2
\][/tex]
5. Reverse the substitution [tex]\( y = x^2 \)[/tex] to find [tex]\( x \)[/tex]:
- For [tex]\( y = 7 \)[/tex]:
[tex]\[
x^2 = 7 \implies x = \pm \sqrt{7}
\][/tex]
- For [tex]\( y = -2 \)[/tex]:
[tex]\[
x^2 = -2 \implies x = \pm \sqrt{-2} = \pm i\sqrt{2}
\][/tex]
6. List all the solutions:
[tex]\[
x = \pm \sqrt{7}, \quad x = \pm i\sqrt{2}
\][/tex]
Therefore, the solutions to the equation [tex]\( x^4 - 5x^2 - 14 = 0 \)[/tex] are:
[tex]\[
x = \pm \sqrt{7} \quad \text{and} \quad x = \pm i \sqrt{2}
\][/tex]
This matches the provided choice:
[tex]\[ \boxed{x = \pm \sqrt{7} \text{ and } x = \pm i \sqrt{2}} \][/tex]
