To find the solutions of the equation [tex]\( x^4 - 5x^2 - 14 = 0 \)[/tex], we can follow a systematic, step-by-step approach.
1. Substitute [tex]\( u = x^2 \)[/tex]:
This substitution reduces the quartic equation to a quadratic form. We substitute [tex]\( u = x^2 \)[/tex], transforming the equation into:
[tex]\[
u^2 - 5u - 14 = 0
\][/tex]
2. Solve the quadratic equation [tex]\( u^2 - 5u - 14 = 0 \)[/tex]:
To solve this quadratic equation, we use the quadratic formula [tex]\( u = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)[/tex], where [tex]\( a = 1 \)[/tex], [tex]\( b = -5 \)[/tex], and [tex]\( c = -14 \)[/tex].
First, we calculate the discriminant:
[tex]\[
\Delta = b^2 - 4ac = (-5)^2 - 4(1)(-14) = 25 + 56 = 81
\][/tex]
Now, we find the roots:
[tex]\[
u_1 = \frac{5 + \sqrt{81}}{2 \cdot 1} = \frac{5 + 9}{2} = \frac{14}{2} = 7
\][/tex]
[tex]\[
u_2 = \frac{5 - \sqrt{81}}{2 \cdot 1} = \frac{5 - 9}{2} = \frac{-4}{2} = -2
\][/tex]
3. Back-substitute [tex]\( u = x^2 \)[/tex]:
We revert back to [tex]\( x \)[/tex] from [tex]\( u \)[/tex] by solving the equations [tex]\( x^2 = u_1 \)[/tex] and [tex]\( x^2 = u_2 \)[/tex].
- For [tex]\( x^2 = 7 \)[/tex]:
[tex]\[
x = \pm \sqrt{7}
\][/tex]
- For [tex]\( x^2 = -2 \)[/tex]:
Since [tex]\( x \)[/tex] must be a complex number to satisfy this equation, we have:
[tex]\[
x = \pm \sqrt{-2} = \pm i \sqrt{2}
\][/tex]
Thus, the solutions to the equation [tex]\( x^4 - 5x^2 - 14 = 0 \)[/tex] are:
[tex]\[
x = \pm \sqrt{7}, \pm i \sqrt{2}
\][/tex]
The correct solution is:
[tex]\[ x= \pm \sqrt{7} \, \text{and} \, x= \pm i \sqrt{2} \][/tex]
