Q:

Find all solutions of the equation algebraically. Check your solutions. (Enter your answers as a comma-separated list x^4-7x^2-144=0

Accepted Solution

A:
Answer:The solutions are: [tex]x=4,\:x=-4,\:x=3i,\:x=-3i[/tex]Step-by-step explanation:Consider the provided equation.[tex]x^4-7x^2-144=0[/tex]Substitute [tex]u=x^2\mathrm{\:and\:}u^2=x^4[/tex][tex]u^2-7u-144=0[/tex][tex]u^2-16u+9u-144=0[/tex][tex](u-16)(u+9)=0[/tex][tex]u=16,\:u=-9[/tex]Substitute back [tex]\:u=x^2[/tex] and solve for x.[tex]x^2=16\\x=\sqrt{16}\\ \quad x=4,\:x=-4[/tex]Or[tex]x^2=-9\\x=\sqrt{-9}\\ \quad x=3i,\:x=-3i[/tex]Hence, the solutions are: [tex]x=4,\:x=-4,\:x=3i,\:x=-3i[/tex]Check:Substitute x=4 in provided equation.[tex]4^4-7(4)^2-144=0[/tex][tex]256-112-144=0[/tex][tex]0=0[/tex]Which is true.Substitute x=-4 in provided equation.[tex](-4)^4-7(-4)^2-144=0[/tex][tex]256-112-144=0[/tex][tex]0=0[/tex]Which is true.Substitute x=3i in provided equation.[tex](3i)^4-7(3i)^2-144=0[/tex][tex]81+63-144=0[/tex][tex]0=0[/tex]Which is true.Substitute x=-3i in provided equation.[tex](-3i)^4-7(-3i)^2-144=0[/tex][tex]81+63-144=0[/tex][tex]0=0[/tex]Which is true.