# Solve for z z^4+10z^2+9=0

z4+10z2+9=0
Substitute u=z2 into the equation. This will make the quadratic formula easy to use.
u2+10u+9=0
u=z2
Factor u2+10u+9 using the AC method.
Consider the form x2+bx+c. Find a pair of integers whose product is c and whose sum is b. In this case, whose product is 9 and whose sum is 10.
1,9
Write the factored form using these integers.
(u+1)(u+9)=0
(u+1)(u+9)=0
If any individual factor on the left side of the equation is equal to 0, the entire expression will be equal to 0.
u+1=0
u+9=0
Set the first factor equal to 0 and solve.
Set the first factor equal to 0.
u+1=0
Subtract 1 from both sides of the equation.
u=-1
u=-1
Set the next factor equal to 0 and solve.
Set the next factor equal to 0.
u+9=0
Subtract 9 from both sides of the equation.
u=-9
u=-9
The final solution is all the values that make (u+1)(u+9)=0 true.
u=-1,-9
Substitute the real value of u=z2 back into the solved equation.
z2=-1
(z2)1=-9
Solve the first equation for z.
z2=-1
Solve the equation for z.
Take the square root of both sides of the equation to eliminate the exponent on the left side.
z=±-1
The complete solution is the result of both the positive and negative portions of the solution.
Rewrite -1 as i.
z=±i
The complete solution is the result of both the positive and negative portions of the solution.
First, use the positive value of the ± to find the first solution.
z=i
Next, use the negative value of the ± to find the second solution.
z=-i
The complete solution is the result of both the positive and negative portions of the solution.
z=i,-i
z=i,-i
z=i,-i
z=i,-i
Solve the second equation for z.
(z2)1=-9
Solve the equation for z.
Take the 1th root of each side of the equation to set up the solution for z
(z2)1⋅11=-91
Remove the perfect root factor z2 under the radical to solve for z.
z2=-91
Take the square root of both sides of the equation to eliminate the exponent on the left side.
z=±-91
The complete solution is the result of both the positive and negative portions of the solution.
Simplify the right side of the equation.
Evaluate -91 as -9.
z=±-9
Rewrite -9 as -1(9).
z=±-1⋅9
Rewrite -1(9) as -1⋅9.
z=±-1⋅9
Rewrite -1 as i.
z=±i⋅9
Rewrite 9 as 32.
z=±i⋅32
Pull terms out from under the radical, assuming positive real numbers.
z=±i⋅3
Move 3 to the left of i.
z=±3i
z=±3i
The complete solution is the result of both the positive and negative portions of the solution.
First, use the positive value of the ± to find the first solution.
z=3i
Next, use the negative value of the ± to find the second solution.
z=-3i
The complete solution is the result of both the positive and negative portions of the solution.
z=3i,-3i
z=3i,-3i
z=3i,-3i
z=3i,-3i
The solution to z4+10z2+9=0 is z=i,-i,3i,-3i.
z=i,-i,3i,-3i
Solve for z z^4+10z^2+9=0

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