Question 737342
Just do exactly as is directed; do what the instructions tell you.  Just that simple.


GIVEN: {{{x^4-5x^2-14=0}}}


a. Let u=x^2, substitute the variables, and write the new equation.

So, do that: {{{u^2-5u-14=0}}}.


 b. Now factor the new equation, apply the zero-product principle, & solve for u.
 
You are told to factor the polynomial expression containing u.
{{{(u +2)(u -7)=0}}}
One of those binomials or the other must be zero, so u has two solutions. 
If {{{u+2=0}}} then {{{u=-2}}}.
If {{{u-7=0}}} then {{{u=+7}}}.



c. Now substitute u=x^2 to solve for x.
Solving for u is part of the solution process but not yet at the finish of the solution.  You want solutions for x.  You remember, the substitution must now be made for u, in the other direction.  x^2=u, because that is how you chose u.


From {{{u=-2}}}, {{{x^2=-2}}}.  If you do not yet know imaginary numbers, then this means no real solution for x.  If you DO know about imaginary numbers, then {{{x=+- sqrt(-2)}}}; or {{{x=-i*sqrt(2)}}} OR {{{x=i*sqrt(2)}}}.


From {{{u=+7}}}, {{{x^2=7}}}.  {{{x=-sqrt(7)}}} OR {{{x=sqrt(7)}}}.


SUMMARY OF THE SOLUTIONS:
{{{highlight(x=-i*sqrt(2))}}} OR {{{highlight(x=i*sqrt(2))}}} OR {{{highlight(x=-sqrt(7))}}} OR {{{highlight(x=sqrt(7))}}}.